Limit of a function

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Limit_of_a_function

See also: Limit of a sequence and Limit point

 

$x$	$\frac{\sin x}{x}$
1	0.841471...
0.1	0.998334...
0.01	0.999983...

Although the function (sin x)/x is not defined at zero, as x becomes closer and closer to zero, (sin x)/x becomes arbitrarily close to 1. In other words, the limit of (sin x)/x, as x approaches zero, equals 1.

In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input.

Formal definitions, first devised in the early 19th century, are given below. Informally, a function f assigns an output f(x) to every input x. We say that the function has a limit L at an input p, if f(x) gets closer and closer to L as x moves closer and closer to p. More specifically, when f is applied to any input sufficiently close to p, the output value is forced arbitrarily close to L. On the other hand, if some inputs very close to p are taken to outputs that stay a fixed distance apart, then we say the limit does not exist.

The notion of a limit has many applications in modern calculus. In particular, the many definitions of continuity employ the concept of limit: roughly, a function is continuous if all of its limits agree with the values of the function. The concept of limit also appears in the definition of the derivative: in the calculus of one variable, this is the limiting value of the slope of secant lines to the graph of a function.

History

Although implicit in the development of calculus of the 17th and 18th centuries, the modern idea of the limit of a function goes back to Bolzano who, in 1817, introduced the basics of the epsilon-delta technique to define continuous functions. However, his work was not known during his lifetime.

In his 1821 book Cours d'analyse, Augustin-Louis Cauchy discussed variable quantities, infinitesimals and limits, and defined continuity of $y=f(x)$ by saying that an infinitesimal change in x necessarily produces an infinitesimal change in y, while (Grabiner 1983) claims that he used a rigorous epsilon-delta definition in proofs. In 1861, Weierstrass first introduced the epsilon-delta definition of limit in the form it is usually written today. He also introduced the notations $lim$ and $\underset{x\to x_0}{lim}{\displaystyle }$.

The modern notation of placing the arrow below the limit symbol is due to Hardy, which is introduced in his book A Course of Pure Mathematics in 1908.

Motivation

Imagine a person walking on a landscape represented by the graph y = f(x). Their horizontal position is given by x, much like the position given by a map of the land or by a global positioning system. Their altitude is given by the coordinate y. Suppose they walk towards a position x = p, as they get closer and closer to this point, they will notice that their altitude approaches a specific value L. If asked about the altitude corresponding to x = p, they would reply by saying y = L.

What, then, does it mean to say, their altitude is approaching L? It means that their altitude gets nearer and nearer to L—except for a possible small error in accuracy. For example, suppose we set a particular accuracy goal for our traveler: they must get within ten meters of L. They report back that indeed, they can get within ten vertical meters of L, arguing that as long as they are within fifty horizontal meters of p, their altitude is always within ten meters of L.

The accuracy goal is then changed: can they get within one vertical meter? Yes, supposing that they are able to move within five horizontal meters of p, their altitude will always remain within one meter from the target altitude L. Summarizing the aforementioned concept we can say that the traveler's altitude approaches L as their horizontal position approaches p, so as to say that for every target accuracy goal, however small it may be, there is some neighbourhood of p whose altitude fulfills that accuracy goal.

The initial informal statement can now be explicated:

The limit of a function f(x) as x approaches p is a number L with the following property: given any target distance from L, there is a distance from p within which the values of f(x) remain within the target distance.

In fact, this explicit statement is quite close to the formal definition of the limit of a function, with values in a topological space.

More specifically, to say that

$\underset{x\to p}{lim}f(x)=L$,

is to say that ƒ(x) can be made as close to L as desired, by making x close enough, but not equal, to p.

The following definitions, known as (ε, δ)-definitions, are the generally accepted definitions for the limit of a function in various contexts.

Functions of a single variable

(ε, δ)-definition of limit

Suppose ${\displaystyle f:\mathbb{R}\to \mathbb{R}}$ is a function defined on the real line, and there are two real numbers p and L. One would say that the limit of f, as x approaches p, is L and written

${\displaystyle \underset{x\to p}{lim}f(x)=L}$,

or alternatively, say $\mathit{f}\mathbf{(}\mathit{x}\mathbf{)}$ tends to $\mathit{L}$ as $\mathit{x}$ tends to $\mathit{p}$, and written:

${\displaystyle f(x)\to L\text{as}x\to p}$,

if the following property holds: for every real ε > 0, there exists a real δ > 0 such that for all real x, 0 < |x − p| < δ implies |f(x) − L| < ε. Symbolically,

${\displaystyle (\mathrm{\forall }\epsilon >0)(\mathrm{\exists }\delta >0)(\mathrm{\forall }x\in \mathbb{R})(0<|x-p|<\delta ⟹|f(x)-L|<\epsilon )}$.

For example, we may say

${\displaystyle \underset{x\to 2}{lim}(4x+1)=9}$

because for every real ε > 0, we can take δ = ε/4, so that for all real x, if 0 < |x − 2| < δ, then |4x + 1 − 9| < ε.

A more general definition applies for functions defined on subsets of the real line. Let (a, b) be an open interval in $\mathbb{R}$, and a number p in (a, b). Let ${\displaystyle f:S\to \mathbb{R}}$ be a real-valued function defined on S — a set that contains all of (a, b), except possibly at p itself. It is then said that the limit of f as x approaches p is L, if:

For every real ε > 0, there exists a real δ > 0 such that for all x ∈ (a, b), 0 < |x − p| < δ implies that |f(x) − L| < ε.

Or, symbolically:

${\displaystyle (\mathrm{\forall }\epsilon >0)(\mathrm{\exists }\delta >0)(\mathrm{\forall }x\in (a,b))(0<|x-p|<\delta ⟹|f(x)-L|<\epsilon )}$.

For example, we may say

${\displaystyle \underset{x\to 1}{lim}\sqrt{x+3}=2}$

because for every real ε > 0, we can take δ = ε, so that for all real x ≥ −3, if 0 < |x − 1| < δ, then |f(x) − 2| < ε. In this example, S = [−3, ∞) contains open intervals around the point 1 (for example, the interval (0, 2)).

Here, note that the value of the limit does not depend on f being defined at p, nor on the value f(p)—if it is defined. For example,

${\displaystyle \underset{x\to 1}{lim}\frac{2x^2-x-1}{x-1}=3}$

because for every ε > 0, we can take δ = ε/2, so that for all real x ≠ 1, if 0 < |x − 1| < δ, then |f(x) − 3| < ε. Note that here f(1) is undefined.

The letters ε and δ can be understood as "error" and "distance". In fact, Cauchy used ε as an abbreviation for "error" in some of his work, though in his definition of continuity, he used an infinitesimal $\alpha$ rather than either ε or δ (see Cours d'Analyse). In these terms, the error (ε) in the measurement of the value at the limit can be made as small as desired, by reducing the distance (δ) to the limit point. As discussed below, this definition also works for functions in a more general context. The idea that δ and ε represent distances helps suggest these generalizations.

Existence and one-sided limits

Main article: One-sided limit

The limit as: x → x₀⁺ ≠ x → x₀⁻. Therefore, the limit as x → x₀ does not exist.

Alternatively, x may approach p from above (right) or below (left), in which case the limits may be written as

$\underset{x\to p^{+}}{lim}f(x)=L$

or

$\underset{x\to p^{-}}{lim}f(x)=L$

respectively. If these limits exist at p and are equal there, then this can be referred to as the limit of f(x) at p. If the one-sided limits exist at p, but are unequal, then there is no limit at p (i.e., the limit at p does not exist). If either one-sided limit does not exist at p, then the limit at p also does not exist.

A formal definition is as follows. The limit of f as x approaches p from above is L if:

For every ε > 0, there exists a δ > 0 such that whenever 0 < x − p < δ, we have |f(x) − L| < ε.

${\displaystyle (\mathrm{\forall }\epsilon >0)(\mathrm{\exists }\delta >0)(\mathrm{\forall }x\in (a,b))(0<x-p<\delta ⟹|f(x)-L|<\epsilon )}$.

The limit of f as x approaches p from below is L if:

For every ε > 0, there exists a δ > 0 such that whenever 0 < p − x < δ, we have |f(x) − L| < ε.

${\displaystyle (\mathrm{\forall }\epsilon >0)(\mathrm{\exists }\delta >0)(\mathrm{\forall }x\in (a,b))(0<p-x<\delta ⟹|f(x)-L|<\epsilon )}$.

If the limit does not exist, then the oscillation of f at p is non-zero.

More general subsets

Apart from open intervals, limits can be defined for functions on arbitrary subsets of R, as follows (Bartle & Sherbert 2000): let ${\displaystyle f:S\to \mathbb{R}}$ be a real-valued function defined on arbitrary ${\displaystyle S\subseteq \mathbb{R}}$. Let p be a limit point of S—that is, p is the limit of some sequence of elements of S distinct from p. Then we say the limit of f, as x approaches p from values in S, is L, written

${\displaystyle \underset{\begin{array}{c}x\to p\\ x\in S\end{array}}{lim}f(x)=L}$

if the following holds:

For every ε > 0, there exists a δ > 0 such that for all x ∈ S, 0 < |x − p| < δ implies that |f(x) − L| < ε.

${\displaystyle (\mathrm{\forall }\epsilon >0)(\mathrm{\exists }\delta >0)(\mathrm{\forall }x\in S)(0<|x-p|<\delta ⟹|f(x)-L|<\epsilon )}$.

The condition that f be defined on S is that S be a subset of the domain of f. This generalization includes as special cases limits on an interval, as well as left-handed limits of real-valued functions (e.g., by taking S to be an open interval of the form $(-\mathrm{\infty },a)$), and right-handed limits (e.g., by taking S to be an open interval of the form $(a,\mathrm{\infty })$). It also extends the notion of one-sided limits to the included endpoints of (half-)closed intervals, so the square root function f(x) = √x can have limit 0 as x approaches 0 from above:

${\displaystyle \underset{\begin{array}{c}x\to 0\\ x\in [0,\mathrm{\infty })\end{array}}{lim}\sqrt{x}=0}$

since for every ε > 0, we may take δ = ε such that for all x ≥ 0, if 0 < |x − 0| < δ, then |f(x) − 0| < ε.

Deleted versus non-deleted limits

The definition of limit given here does not depend on how (or whether) f is defined at p. Bartle (1967) refers to this as a deleted limit, because it excludes the value of f at p. The corresponding non-deleted limit does depend on the value of f at p, if p is in the domain of f. Let ${\displaystyle f:S\to \mathbb{R}}$ be a real-valued function. The non-deleted limit of f, as x approaches p, is L if

For every ε > 0, there exists a δ > 0 such that for all x ∈ S, |x − p| < δ implies |f(x) − L| < ε

${\displaystyle (\mathrm{\forall }\epsilon >0)(\mathrm{\exists }\delta >0)(\mathrm{\forall }x\in S)(|x-p|<\delta ⟹|f(x)-L|<\epsilon )}$.

The definition is the same, except that the neighborhood |x − p| < δ now includes the point p, in contrast to the deleted neighborhood 0 < |x − p| < δ. This makes the definition of a non-deleted limit less general. One of the advantages of working with non-deleted limits is that they allow to state the theorem about limits of compositions without any constraints on the functions (other than the existence of their non-deleted limits) (Hubbard (2015)).

Bartle (1967) notes that although by "limit" some authors do mean this non-deleted limit, deleted limits are the most popular. For example, Apostol (1974), Courant (1924), Hardy (1921), Rudin (1964), Whittaker & Watson (1902) all take "limit" to mean the deleted limit.

Examples

Non-existence of one-sided limit(s)

The function without a limit, at an essential discontinuity

The function

${\displaystyle f(x)=\{\begin{array}{ll}\sin \frac{5}{x-1}& \text{ for }x<1\\ 0& \text{ for }x=1\\ \frac{0.1}{x-1}& \text{ for }x>1\end{array}}$

has no limit at $x_0=1$ (the left-hand limit does not exist due to the oscillatory nature of the sine function, and the right-hand limit does not exist due to the asymptotic behaviour of the reciprocal function), but has a limit at every other x-coordinate.

The function

${\displaystyle f(x)=\{\begin{array}{ll}1& x\text{ rational }\\ 0& x\text{ irrational }\end{array}}$

(a.k.a., the Dirichlet function) has no limit at any x-coordinate.

Non-equality of one-sided limits

The function

${\displaystyle f(x)=\{\begin{array}{ll}1& \text{ for }x<0\\ 2& \text{ for }x\ge 0\end{array}}$

has a limit at every non-zero x-coordinate (the limit equals 1 for negative x and equals 2 for positive x). The limit at x = 0 does not exist (the left-hand limit equals 1, whereas the right-hand limit equals 2).

Limits at only one point

The functions

${\displaystyle f(x)=\{\begin{array}{ll}x& x\text{ rational }\\ 0& x\text{ irrational }\end{array}}$

and

${\displaystyle f(x)=\{\begin{array}{ll}|x|& x\text{ rational }\\ 0& x\text{ irrational }\end{array}}$

both have a limit at x = 0 and it equals 0.

Limits at countably many points

The function

${\displaystyle f(x)=\{\begin{array}{ll}\sin x& x\text{ irrational }\\ 1& x\text{ rational }\end{array}}$

has a limit at any x-coordinate of the form $\frac{\pi }{2}+2n\pi$, where n is any integer.

Limits involving infinity

Limits at infinity

The limit of this function at infinity exists

Let ${\displaystyle f:S\to \mathbb{R}}$ be a function defined on ${\displaystyle S\subseteq \mathbb{R}}$. The limit of f as x approaches infinity is L, denoted

${\displaystyle \underset{x\to \mathrm{\infty }}{lim}f(x)=L}$,

means that:

For every ε > 0, there exists a c > 0 such that whenever x > c, we have |f(x) − L| < ε.

${\displaystyle (\mathrm{\forall }\epsilon >0)(\mathrm{\exists }c>0)(\mathrm{\forall }x\in S)(x>c⟹|f(x)-L|<\epsilon )}$.

Similarly, the limit of f as x approaches minus infinity is L, denoted

${\displaystyle \underset{x\to -\mathrm{\infty }}{lim}f(x)=L}$,

means that:

For every ε > 0, there exists a c > 0 such that whenever x < −c, we have |f(x) − L| < ε.

${\displaystyle (\mathrm{\forall }\epsilon >0)(\mathrm{\exists }c>0)(\mathrm{\forall }x\in S)(x<-c⟹|f(x)-L|<\epsilon )}$.

For example,

${\displaystyle \underset{x\to \mathrm{\infty }}{lim}\left(-\frac{3\sin x}{x}+4\right)=4}$

because for every ε > 0, we can take c = 3/ε such that for all real x, if x > c, then |f(x) − 4| < ε.

Another example is that

${\displaystyle \underset{x\to -\mathrm{\infty }}{lim}{e}^x=0}$

because for every ε > 0, we can take c = max{1, −ln(ε)} such that for all real x, if x < −c, then |f(x) − 0| < ε.

Infinite limits

For a function whose values grow without bound, the function diverges and the usual limit does not exist. However, in this case one may introduce limits with infinite values.

Let ${\displaystyle f:S\to \mathbb{R}}$ be a function defined on ${\displaystyle S\subseteq \mathbb{R}}$. The statement the limit of f as x approaches p is infinity, denoted

${\displaystyle \underset{x\to p}{lim}f(x)=\mathrm{\infty },}$

means that:

For every N > 0, there exists a δ > 0 such that whenever 0 < |x − p| < δ, we have f(x) > N.

${\displaystyle (\mathrm{\forall }N>0)(\mathrm{\exists }\delta >0)(\mathrm{\forall }x\in S)(0<|x-p|<\delta ⟹f(x)>N)}$.

The statement the limit of f as x approaches p is minus infinity, denoted

${\displaystyle \underset{x\to p}{lim}f(x)=-\mathrm{\infty },}$

means that:

For every N > 0, there exists a δ > 0 such that whenever 0 < |x − p| < δ, we have f(x) < −N.

${\displaystyle (\mathrm{\forall }N>0)(\mathrm{\exists }\delta >0)(\mathrm{\forall }x\in S)(0<|x-p|<\delta ⟹f(x)<-N)}$.

For example,

${\displaystyle \underset{x\to 1}{lim}\frac{1}{(x-1{)}^2}=\mathrm{\infty }}$

because for every N > 0, we can take $\delta =1/\sqrt{N}$δ = 1/√N such that for all real x > 0, if 0 < x − 1 < δ, then f(x) > N.

These ideas can be used together to produce definitions for different combinations, such as

${\displaystyle \underset{x\to \mathrm{\infty }}{lim}f(x)=\mathrm{\infty }}$, or ${\displaystyle \underset{x\to p^{+}}{lim}f(x)=-\mathrm{\infty }}$.

For example,

${\displaystyle \underset{x\to 0^{+}}{lim}\ln x=-\mathrm{\infty }}$

because for every N > 0, we can take δ = e^−N such that for all real x > 0, if 0 < x − 0 < δ, then f(x) < −N.

Limits involving infinity are connected with the concept of asymptotes.

These notions of a limit attempt to provide a metric space interpretation to limits at infinity. In fact, they are consistent with the topological space definition of limit if

• a neighborhood of −∞ is defined to contain an interval [−∞, c) for some c ∈ R,
• a neighborhood of ∞ is defined to contain an interval (c, ∞] where c ∈ R, and
• a neighborhood of a ∈ R is defined in the normal way metric space R.

In this case, R is a topological space and any function of the form f: X → Y with X, Y⊆ R is subject to the topological definition of a limit. Note that with this topological definition, it is easy to define infinite limits at finite points, which have not been defined above in the metric sense.

Alternative notation

Many authors allow for the projectively extended real line to be used as a way to include infinite values as well as extended real line. With this notation, the extended real line is given as R ∪ {−∞, +∞} and the projectively extended real line is R ∪ {∞} where a neighborhood of ∞ is a set of the form {x: |x| > c}. The advantage is that one only needs three definitions for limits (left, right, and central) to cover all the cases. As presented above, for a completely rigorous account, we would need to consider 15 separate cases for each combination of infinities (five directions: −∞, left, central, right, and +∞; three bounds: −∞, finite, or +∞). There are also noteworthy pitfalls. For example, when working with the extended real line, $x^{-1}$ does not possess a central limit (which is normal):

${\displaystyle \underset{x\to 0^{+}}{lim}\frac{1}{x}=+\mathrm{\infty },\underset{x\to 0^{-}}{lim}\frac{1}{x}=-\mathrm{\infty }}$.

In contrast, when working with the projective real line, infinities (much like 0) are unsigned, so, the central limit does exist in that context:

${\displaystyle \underset{x\to 0^{+}}{lim}\frac{1}{x}=\underset{x\to 0^{-}}{lim}\frac{1}{x}=\underset{x\to 0}{lim}\frac{1}{x}=\mathrm{\infty }}$.

In fact there are a plethora of conflicting formal systems in use. In certain applications of numerical differentiation and integration, it is, for example, convenient to have signed zeroes. A simple reason has to do with the converse of $\underset{x\to 0^{-}}{lim}{x}^{-1}=-\mathrm{\infty }$, namely, it is convenient for $\underset{x\to -\mathrm{\infty }}{lim}{x}^{-1}=-0$ to be considered true. Such zeroes can be seen as an approximation to infinitesimals.

Limits at infinity for rational functions

Horizontal asymptote about y = 4

There are three basic rules for evaluating limits at infinity for a rational function f(x) = p(x)/q(x): (where p and q are polynomials):

• If the degree of p is greater than the degree of q, then the limit is positive or negative infinity depending on the signs of the leading coefficients;
• If the degree of p and q are equal, the limit is the leading coefficient of p divided by the leading coefficient of q;
• If the degree of p is less than the degree of q, the limit is 0.

If the limit at infinity exists, it represents a horizontal asymptote at y = L. Polynomials do not have horizontal asymptotes; such asymptotes may however occur with rational functions.

Functions of more than one variable

Ordinary limits

By noting that |x − p| represents a distance, the definition of a limit can be extended to functions of more than one variable. In the case of a function ${\displaystyle f:S\times T\to \mathbb{R}}$ defined on ${\displaystyle S\times T\subseteq {\mathbb{R}}^2}$, we defined the limit as follows: the limit of f as (x, y) approaches (p, q) is L, written

$\underset{(x,y)\to (p,q)}{lim}f(x,y)=L$

if the following condition holds:

For every ε > 0, there exists a δ > 0 such that for all x in S and y in T, whenever 0 < √(x−p)² + (y−q)² < δ, we have |f(x, y) − L| < ε.

${\displaystyle (\mathrm{\forall }\epsilon >0)(\mathrm{\exists }\delta >0)(\mathrm{\forall }x\in S)(\mathrm{\forall }y\in T)(0<\sqrt{(x-p{)}^2+(y-q{)}^2}<\delta ⟹|f(x,y)-L|<\epsilon )}$.

Here √(x−p)² + (y−q)² is the Euclidean distance between (x, y) and (p, q). (This can in fact be replaced by any norm ||(x, y) − (p, q)||, and be extended to any number of variables.)

For example, we may say

${\displaystyle \underset{(x,y)\to (0,0)}{lim}\frac{x^4}{x^2+y^2}=0}$

because for every ε > 0, we can take δ = √ε such that for all real x ≠ 0 and real y ≠ 0, if 0 < √(x−0)² + (y−0)² < δ, then |f(x, y) − 0| < ε.

Similar to the case in single variable, the value of f at (p, q) does not matter in this definition of limit.

For such a multivariable limit to exist, this definition requires the value of f approaches L along every possible path approaching (p, q).</ref> In the above example, the function

${\displaystyle f(x,y)=\frac{x^4}{x^2+y^2}}$

satisfies this condition. This can be seen by considering the polar coordinates (x, y) = (r cos(θ), r sin(θ)) → (0, 0), which gives

${\displaystyle \underset{r\to 0}{lim}f(r\cos \theta ,r\sin \theta )=\underset{r\to 0}{lim}\frac{r^4{\cos }^4\theta }{r^2}=\underset{r\to 0}{lim}{r}^2{\cos }^4\theta }$.

Here θ = θ(r) is a function of r which controls the shape of the path along which f is approaching (p, q). Since cos(θ) is bounded between [−1, 1], by the sandwich theorem, this limit tends to 0.

In contrast, the function

${\displaystyle f(x,y)=\frac{xy}{x^2+y^2}}$

does not have a limit at (0, 0). Taking the path (x, y) = (t, 0) → (0, 0), we obtain

${\displaystyle \underset{t\to 0}{lim}f(t,0)=\underset{t\to 0}{lim}\frac{0}{t^2}=0}$,

while taking the path (x, y) = (t, t) → (0, 0), we obtain

${\displaystyle \underset{t\to 0}{lim}f(t,t)=\underset{t\to 0}{lim}\frac{t^2}{t^2+t^2}=\frac{1}{2}}$.

Since the two values do not agree, f does not tend to a single value as (x, y) approaches (0, 0).

Multiple limits

Although less commonly used, there is another type of limit for a multivariable function, known as the multiple limit. For a two-variable function, this is the double limit. Let ${\displaystyle f:S\times T\to \mathbb{R}}$ be defined on ${\displaystyle S\times T\subseteq {\mathbb{R}}^2}$, we say the double limit of f as x approaches p and y approaches q is L, written

${\displaystyle \underset{\begin{array}{c}x\to p\\ y\to q\end{array}}{lim}f(x,y)=L}$

if the following condition holds:

For every ε > 0, there exists a δ > 0 such that for all x in S and y in T, whenever 0 < |x − p| < δ and 0 < |y−q| < δ, we have |f(x, y) − L| < ε.

${\displaystyle (\mathrm{\forall }\epsilon >0)(\mathrm{\exists }\delta >0)(\mathrm{\forall }x\in S)(\mathrm{\forall }y\in T)((0<|x-p|<\delta )\wedge (0<|y-q|<\delta )⟹|f(x,y)-L|<\epsilon )}$.

For such a double limit to exist, this definition requires the value of f approaches L along every possible path approaching (p, q), excluding the two lines x = p and y = q. As a result, the multiple limit is a weaker notion than the ordinary limit: if the ordinary limit exists and equals L, then the multiple limit exists and also equals L. The converse is not true: the existence of the multiple limits does not imply the existence of the ordinary limit. Consider the example

${\displaystyle f(x,y)=\{\begin{array}{l}1\text{for}xy\ne 0\\ 0\text{for}xy=0\end{array}}$

where

${\displaystyle \underset{\begin{array}{c}x\to 0\\ y\to 0\end{array}}{lim}f(x,y)=1}$

but

${\displaystyle \underset{(x,y)\to (0,0)}{lim}f(x,y)}$ does not exist.

If the domain of f is restricted to ${\displaystyle (S\setminus \{p\})\times (T\setminus \{q\})}$, then the two definitions of limits coincide.

Multiple limits at infinity

The concept of multiple limit can extend to the limit at infinity, in a way similar to that of a single variable function. For ${\displaystyle f:S\times T\to \mathbb{R}}$, we say the double limit of f as x and y approaches infinity is L, written

${\displaystyle \underset{\begin{array}{c}x\to \mathrm{\infty }\\ y\to \mathrm{\infty }\end{array}}{lim}f(x,y)=L}$

if the following condition holds:

For every ε > 0, there exists a c > 0 such that for all x in S and y in T, whenever x > c and y > c, we have |f(x, y) − L| < ε.

${\displaystyle (\mathrm{\forall }\epsilon >0)(\mathrm{\exists }c>0)(\mathrm{\forall }x\in S)(\mathrm{\forall }y\in T)((x>c)\wedge (y>c)⟹|f(x,y)-L|<\epsilon )}$.

We say the double limit of f as x and y approaches minus infinity is L, written

${\displaystyle \underset{\begin{array}{c}x\to -\mathrm{\infty }\\ y\to -\mathrm{\infty }\end{array}}{lim}f(x,y)=L}$

if the following condition holds:

For every ε > 0, there exists a c > 0 such that x in S and y in T, whenever x < −c and y < −c, we have |f(x, y) − L| < ε.

${\displaystyle (\mathrm{\forall }\epsilon >0)(\mathrm{\exists }c>0)(\mathrm{\forall }x\in S)(\mathrm{\forall }y\in T)((x<-c)\wedge (y<-c)⟹|f(x,y)-L|<\epsilon )}$.

Pointwise limits and uniform limits

Main articles: Pointwise convergence and Uniform convergence

Let ${\displaystyle f:S\times T\to \mathbb{R}}$. Instead of taking limit as (x, y) → (p, q), we may consider taking the limit of just one variable, say, x → p, to obtain a single-variable function of y, namely ${\displaystyle g:T\to \mathbb{R}}$. In fact, this limiting process can be done in two distinct ways. The first one is called pointwise limit. We say the pointwise limit of f as x approaches p is g, denoted

${\displaystyle \underset{x\to p}{lim}f(x,y)=g(y)}$, or

${\displaystyle \underset{x\to p}{lim}f(x,y)=g(y)\text{pointwise}}$.

Alternatively, we may say f tends to g pointwise as x approaches p, denoted

${\displaystyle f(x,y)\to g(y)\text{as}x\to p}$, or

${\displaystyle f(x,y)\to g(y)\text{pointwise}\text{as}x\to p}$.

This limit exists if the following holds:

For every ε > 0 and every fixed y in T, there exists a δ(ε, y) > 0 such that for all x in S, whenever 0 < |x − p| < δ, we have |f(x, y) − g(y)| < ε.

${\displaystyle (\mathrm{\forall }\epsilon >0)(\mathrm{\forall }y\in T)(\mathrm{\exists }\delta >0)(\mathrm{\forall }x\in S)(0<|x-p|<\delta ⟹|f(x,y)-g(y)|<\epsilon )}$.

Here, δ = δ(ε, y) is a function of both ε and y. Each δ is chosen for a specific point of y. Hence we say the limit is pointwise in y. For example,

${\displaystyle f(x,y)=\frac{x}{\cos y}}$

has a pointwise limit of constant zero function

${\displaystyle \underset{x\to 0}{lim}f(x,y)=0(y)\text{pointwise}}$

because for every fixed y, the limit is clearly 0. This argument fails if y is not fixed: if y is very close to π/2, the value of the fraction may deviate from 0.

This leads to another definition of limit, namely the uniform limit. We say the uniform limit of f on T as x approaches p is g, denoted

${\displaystyle \underset{\begin{array}{c}x\to p\\ y\in T\end{array}}{\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{f}lim}f(x,y)=g(y)}$, or

${\displaystyle \underset{x\to p}{lim}f(x,y)=g(y)\text{uniformly on}T}$.

Alternatively, we may say f tends to g uniformly on T as x approaches p, denoted

${\displaystyle f(x,y)\rightrightarrows g(y)\text{on}T\text{as}x\to p}$, or

${\displaystyle f(x,y)\to g(y)\text{uniformly on}T\text{as}x\to p}$.

This limit exists if the following holds:

For every ε > 0, there exists a δ(ε) > 0 such that for all x in S and y in T, whenever 0 < |x − p| < δ, we have |f(x, y) − g(y)| < ε.

${\displaystyle (\mathrm{\forall }\epsilon >0)(\mathrm{\exists }\delta >0)(\mathrm{\forall }x\in S)(\mathrm{\forall }y\in T)(0<|x-p|<\delta ⟹|f(x,y)-g(y)|<\epsilon )}$.

Here, δ = δ(ε) is a function of only ε but not y. In other words, δ is uniformly applicable to all y in T. Hence we say the limit is uniform in y. For example,

${\displaystyle f(x,y)=x\cos y}$

has a uniform limit of constant zero function

${\displaystyle \underset{x\to 0}{lim}f(x,y)=0(y)\text{ uniformly on}\mathbb{R}}$

because for all real y, cos(y) is bounded between [−1, 1]. Hence no matter how y behaves, we may use the sandwich theorem to show that the limit is 0.

Iterated limits

Main article: Iterated limits

Let ${\displaystyle f:S\times T\to \mathbb{R}}$. We may consider taking the limit of just one variable, say, x → p, to obtain a single-variable function of y, namely ${\displaystyle g:T\to \mathbb{R}}$, and then take limit in the other variable, namely y → q, to get a number $L$. Symbolically,

${\displaystyle \underset{y\to q}{lim}\underset{x\to p}{lim}f(x,y)=\underset{y\to q}{lim}g(y)=L}$.

This limit is known as iterated limit of the multivariable function. The order of taking limits may affect the result, i.e.,

${\displaystyle \underset{y\to q}{lim}\underset{x\to p}{lim}f(x,y)\ne \underset{x\to p}{lim}\underset{y\to q}{lim}f(x,y)}$ in general.

A sufficient condition of equality is given by the Moore-Osgood theorem, which requires the limit ${\displaystyle \underset{x\to p}{lim}f(x,y)=g(y)}$ to be uniform on T.

Functions on metric spaces

Suppose M and N are subsets of metric spaces A and B, respectively, and f : M → N is defined between M and N, with x ∈ M, p a limit point of M and L ∈ N. It is said that the limit of f as x approaches p is L and write

$\underset{x\to p}{lim}f(x)=L$

if the following property holds:

For every ε > 0, there exists a δ > 0 such that for all points x ∈ M, 0 < d_A(x, p) < δ implies d_B(f(x), L) < ε.

${\displaystyle (\mathrm{\forall }\epsilon >0)(\mathrm{\exists }\delta >0)(\mathrm{\forall }x\in M)(0<d_A(x,p)<\delta ⟹d_B(f(x),L)<\epsilon )}$.

Again, note that p need not be in the domain of f, nor does L need to be in the range of f, and even if f(p) is defined it need not be equal to L.

Euclidean metric

The limit in Euclidean space is a direct generalization of limits to vector-valued functions. For example, we may consider a function ${\displaystyle f:S\times T\to {\mathbb{R}}^3}$ such that

${\displaystyle f(x,y)=(f_1(x,y),f_2(x,y),f_3(x,y))}$.

Then, under the usual Euclidean metric,

${\displaystyle \underset{(x,y)\to (p,q)}{lim}f(x,y)=(L_1,L_2,L_3)}$

if the following holds:

For every ε > 0, there exists a δ > 0 such that for all x in S and y in T, 0 < √(x−p)² + (y−q)² < δ implies √(f₁−L₁)² + (f₂−L₂)² + (f₃−L₃)² < ε.

${\displaystyle (\mathrm{\forall }\epsilon >0)(\mathrm{\exists }\delta >0)(\mathrm{\forall }x\in S)(\mathrm{\forall }y\in T)(0<\sqrt{(x-p{)}^2+(y-q{)}^2}<\delta ⟹\sqrt{(f_1-L_1{)}^2+(f_2-L_2{)}^2+(f_3-L_3{)}^2}<\epsilon )}$.

In this example, the function concerned are finite-dimension vector-valued function. In this case, the limit theorem for vector-valued function states that if the limit of each component exists, then the limit of a vector-valued function equals the vector with each component taken the limit:

${\displaystyle \underset{(x,y)\to (p,q)}{lim}\left(f_1(x,y),f_2(x,y),f_3(x,y)\right)=\left(\underset{(x,y)\to (p,q)}{lim}{f}_1(x,y),\underset{(x,y)\to (p,q)}{lim}{f}_2(x,y),\underset{(x,y)\to (p,q)}{lim}{f}_3(x,y)\right)}$.

Manhattan metric

One might also want to consider spaces other than Euclidean space. An example would be the Manhattan space. Consider ${\displaystyle f:S\to {\mathbb{R}}^2}$ such that

${\displaystyle f(x)=(f_1(x),f_2(x))}$.

Then, under the Manhattan metric,

${\displaystyle \underset{x\to p}{lim}f(x)=(L_1,L_2)}$

if the following holds:

For every ε > 0, there exists a δ > 0 such that for all x in S, 0 < |x − p| < δ implies |f₁−L₁| + |f₂−L₂| < ε.

${\displaystyle (\mathrm{\forall }\epsilon >0)(\mathrm{\exists }\delta >0)(\mathrm{\forall }x\in S)(0<|x-p|<\delta ⟹|f_1-L_1|+|f_2-L_2|<\epsilon )}$.

Since this is also a finite-dimension vector-valued function, the limit theorem stated above also applies.

Uniform metric

Finally, we will discuss the limit in function space, which has infinite dimensions. Consider a function f(x, y) in the function space ${\displaystyle S\times T\to \mathbb{R}}$. We want to find out as x approaches p, how f(x, y) will tend to another function g(y), which is in the function space ${\displaystyle T\to \mathbb{R}}$. The "closeness" in this function space may be measured under the uniform metric. Then, we will say the uniform limit of f on T as x approaches p is g and write

${\displaystyle \underset{\begin{array}{c}x\to p\\ y\in T\end{array}}{\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{f}lim}f(x,y)=g(y)}$, or

${\displaystyle \underset{x\to p}{lim}f(x,y)=g(y)\text{uniformly on}T}$,

if the following holds:

For every ε > 0, there exists a δ > 0 such that for all x in S, 0 < |x − p| < δ implies sup_y∈T |f(x,y) − g(y)| < ε.

${\displaystyle (\mathrm{\forall }\epsilon >0)(\mathrm{\exists }\delta >0)(\mathrm{\forall }x\in S)(0<|x-p|<\delta ⟹\underset{y\in T}{sup}|f(x,y)-g(y)|<\epsilon )}$.

In fact, one can see that this definition is equivalent to that of the uniform limit of a multivariable function introduced in the previous section.

Functions on topological spaces

See also: Filters in topology § Limits of functions

Suppose X,Y are topological spaces with Y a Hausdorff space. Let p be a limit point of Ω ⊆ X, and L ∈Y. For a function f : Ω → Y, it is said that the limit of f as x approaches p is L, written

$\underset{x\to p}{lim}f(x)=L$,

if the following property holds:

For every open neighborhood V of L, there exists an open neighborhood U of p such that f(U ∩ Ω − {p}) ⊆ V.

This last part of the definition can also be phrased "there exists an open punctured neighbourhood U of p such that f(U∩Ω) ⊆ V ".

The domain of f does not need to contain p. If it does, then the value of f at p is irrelevant to the definition of the limit. In particular, if the domain of f is X − {p} (or all of X), then the limit of f as x → p exists and is equal to L if, for all subsets Ω of X with limit point p, the limit of the restriction of f to Ω exists and is equal to L. Sometimes this criterion is used to establish the non-existence of the two-sided limit of a function on R by showing that the one-sided limits either fail to exist or do not agree. Such a view is fundamental in the field of general topology, where limits and continuity at a point are defined in terms of special families of subsets, called filters, or generalized sequences known as nets.

Alternatively, the requirement that Y be a Hausdorff space can be relaxed to the assumption that Y be a general topological space, but then the limit of a function may not be unique. In particular, one can no longer talk about the limit of a function at a point, but rather a limit or the set of limits at a point.

A function is continuous at a limit point p of and in its domain if and only if f(p) is the (or, in the general case, a) limit of f(x) as x tends to p.

There is another type of limit of a function, namely the sequential limit. Let f : X → Y be a mapping from a topological space X into a Hausdorff space Y, p ∈ X a limit point of X and L ∈ Y. The sequential limit of f as x tends to p is L if

For every sequence (x_n) in X − {p} that converges to p, the sequence f(x_n) converges to L.

If L is the limit (in the sense above) of f as x approaches p, then it is a sequential limit as well, however the converse need not hold in general. If in addition X is metrizable, then L is the sequential limit of f as x approaches p if and only if it is the limit (in the sense above) of f as x approaches p.

Other characterizations

In terms of sequences

For functions on the real line, one way to define the limit of a function is in terms of the limit of sequences. (This definition is usually attributed to Eduard Heine.) In this setting:

$\underset{x\to a}{lim}f(x)=L$

if, and only if, for all sequences $x_n$ (with $x_n$ not equal to a for all n) converging to $a$ the sequence $f(x_n)$ converges to $L$. It was shown by Sierpiński in 1916 that proving the equivalence of this definition and the definition above, requires and is equivalent to a weak form of the axiom of choice. Note that defining what it means for a sequence $x_n$ to converge to $a$ requires the epsilon, delta method.

Similarly as it was the case of Weierstrass's definition, a more general Heine definition applies to functions defined on subsets of the real line. Let f be a real-valued function with the domain Dm(f). Let a be the limit of a sequence of elements of Dm(f) \ {a}. Then the limit (in this sense) of f is L as x approaches p if for every sequence $x_n$ ∈ Dm(f) \ {a} (so that for all n, $x_n$ is not equal to a) that converges to a, the sequence $f(x_n)$ converges to $L$. This is the same as the definition of a sequential limit in the preceding section obtained by regarding the subset Dm(f) of R as a metric space with the induced metric.

In non-standard calculus

In non-standard calculus the limit of a function is defined by:

$\underset{x\to a}{lim}f(x)=L$

if and only if for all $x\in {\mathbb{R}}^{\ast }$, $f^{\ast }(x)-L$ is infinitesimal whenever $x-a$ is infinitesimal. Here ${\mathbb{R}}^{\ast }$ are the hyperreal numbers and $f^{\ast }$ is the natural extension of f to the non-standard real numbers. Keisler proved that such a hyperreal definition of limit reduces the quantifier complexity by two quantifiers. On the other hand, Hrbacek writes that for the definitions to be valid for all hyperreal numbers they must implicitly be grounded in the ε-δ method, and claims that, from the pedagogical point of view, the hope that non-standard calculus could be done without ε-δ methods cannot be realized in full. Bŀaszczyk et al. detail the usefulness of microcontinuity in developing a transparent definition of uniform continuity, and characterize Hrbacek's criticism as a "dubious lament".

In terms of nearness

At the 1908 international congress of mathematics F. Riesz introduced an alternate way defining limits and continuity in concept called "nearness". A point $x$ is defined to be near a set $A\subseteq \mathbb{R}$ if for every $r>0$ there is a point $a\in A$ so that $|x-a|<r$. In this setting the

$\underset{x\to a}{lim}f(x)=L$

if and only if for all $A\subseteq \mathbb{R}$, $L$ is near $f(A)$ whenever $a$ is near $A$. Here $f(A)$ is the set $\{f(x)|x\in A\}$. This definition can also be extended to metric and topological spaces.

Relationship to continuity

Main article: Continuous function

The notion of the limit of a function is very closely related to the concept of continuity. A function ƒ is said to be continuous at c if it is both defined at c and its value at c equals the limit of f as x approaches c:

$\underset{x\to c}{lim}f(x)=f(c).$

(We have here assumed that c is a limit point of the domain of f.)

Properties

If a function f is real-valued, then the limit of f at p is L if and only if both the right-handed limit and left-handed limit of f at p exist and are equal to L.

The function f is continuous at p if and only if the limit of f(x) as x approaches p exists and is equal to f(p). If f : M → N is a function between metric spaces M and N, then it is equivalent that f transforms every sequence in M which converges towards p into a sequence in N which converges towards f(p).

If N is a normed vector space, then the limit operation is linear in the following sense: if the limit of f(x) as x approaches p is L and the limit of g(x) as x approaches p is P, then the limit of f(x) + g(x) as x approaches p is L + P. If a is a scalar from the base field, then the limit of af(x) as x approaches p is aL.

If f and g are real-valued (or complex-valued) functions, then taking the limit of an operation on f(x) and g(x) (e.g., $f+g$, ${\displaystyle f-g}$, $f\times g$, $f/g$, ${\displaystyle f^g}$) under certain conditions is compatible with the operation of limits of f(x) and g(x). This fact is often called the algebraic limit theorem. The main condition needed to apply the following rules is that the limits on the right-hand sides of the equations exist (in other words, these limits are finite values including 0). Additionally, the identity for division requires that the denominator on the right-hand side is non-zero (division by 0 is not defined), and the identity for exponentiation requires that the base is positive, or zero while the exponent is positive (finite).

${\displaystyle \begin{array}{cccc}\underset{x\to p}{lim}& (f(x)+g(x))& =& \underset{x\to p}{lim}f(x)+\underset{x\to p}{lim}g(x)\\ \underset{x\to p}{lim}& (f(x)-g(x))& =& \underset{x\to p}{lim}f(x)-\underset{x\to p}{lim}g(x)\\ \underset{x\to p}{lim}& (f(x)\cdot g(x))& =& \underset{x\to p}{lim}f(x)\cdot \underset{x\to p}{lim}g(x)\\ \underset{x\to p}{lim}& (f(x)/g(x))& =& \underset{x\to p}{lim}f(x)/\underset{x\to p}{lim}g(x)\\ \underset{x\to p}{lim}& f(x{)}^{g(x)}& =& \underset{x\to p}{lim}f(x{)}^{\underset{x\to p}{lim}g(x)}\end{array}}$

These rules are also valid for one-sided limits, including when p is ∞ or −∞. In each rule above, when one of the limits on the right is ∞ or −∞, the limit on the left may sometimes still be determined by the following rules.

• q + ∞ = ∞ if q ≠ −∞
• q × ∞ = ∞ if q > 0
• q × ∞ = −∞ if q < 0
• q / ∞ = 0 if q ≠ ∞ and q ≠ −∞
• ∞^q = 0 if q < 0
• ∞^q = ∞ if q > 0
• q^∞ = 0 if 0 < q < 1
• q^∞ = ∞ if q > 1
• q^−∞ = ∞ if 0 < q < 1
• q^−∞ = 0 if q > 1

(see also Extended real number line).

In other cases the limit on the left may still exist, although the right-hand side, called an indeterminate form, does not allow one to determine the result. This depends on the functions f and g. These indeterminate forms are:

• 0 / 0
• ±∞ / ±∞
• 0 × ±∞
• ∞ + −∞
• 0⁰
• ∞⁰
• 1^±∞

See further L'Hôpital's rule below and Indeterminate form.

Limits of compositions of functions

In general, from knowing that

$\underset{y\to b}{lim}f(y)=c$ and ${\displaystyle \underset{x\to a}{lim}g(x)=b}$,

it does not follow that ${\displaystyle \underset{x\to a}{lim}f(g(x))=c}$. However, this "chain rule" does hold if one of the following additional conditions holds:

• f(b) = c (that is, f is continuous at b), or
• g does not take the value b near a (that is, there exists a $\delta >0$ such that if $0<|x-a|<\delta$ then $|g(x)-b|>0$).

As an example of this phenomenon, consider the following function that violates both additional restrictions:

${\displaystyle f(x)=g(x)=\{\begin{array}{ll}0& \text{if }x\ne 0\\ 1& \text{if }x=0\end{array}.}$

Since the value at f(0) is a removable discontinuity,

$\underset{x\to a}{lim}f(x)=0$ for all $a$.

Thus, the naïve chain rule would suggest that the limit of f(f(x)) is 0. However, it is the case that

$f(f(x))=\{\begin{array}{ll}1& \text{if }x\ne 0\\ 0& \text{if }x=0\end{array}$

and so

$\underset{x\to a}{lim}f(f(x))=1$ for all $a$.

Limits of special interest

Main article: List of limits

Rational functions

For $n$ a nonnegative integer and constants $a_1,a_2,a_3,\dots ,a_n$ and $b_1,b_2,b_3,\dots ,b_n$,

• $\underset{x\to \mathrm{\infty }}{lim}\frac{a_1{x}^n+a_2{x}^{n-1}+a_3{x}^{n-2}+...+a_n}{b_1{x}^n+b_2{x}^{n-1}+b_3{x}^{n-2}+...+b_n}=\frac{a_1}{b_1}$

This can be proven by dividing both the numerator and denominator by $x^n$. If the numerator is a polynomial of higher degree, the limit does not exist. If the denominator is of higher degree, the limit is 0.

Trigonometric functions

• $\underset{x\to 0}{lim}\frac{\sin x}{x}=1$
• $\underset{x\to 0}{lim}\frac{1-\cos x}{x}=0$

Exponential functions

• ${\displaystyle \underset{x\to 0}{lim}(1+x{)}^{\frac{1}{x}}=\underset{r\to \mathrm{\infty }}{lim}{\left(1+\frac{1}{r}\right)}^r=e}$
• $\underset{x\to 0}{lim}\frac{e^x-1}{x}=1$
• $\underset{x\to 0}{lim}\frac{e^{ax}-1}{bx}=\frac{a}{b}$
• $\underset{x\to 0}{lim}\frac{c^{ax}-1}{bx}=\frac{a}{b}\ln c$
• ${\displaystyle \underset{x\to 0^{+}}{lim}{x}^x=1}$

Logarithmic functions

• $\underset{x\to 0}{lim}\frac{\ln (1+x)}{x}=1$
• $\underset{x\to 0}{lim}\frac{\ln (1+ax)}{bx}=\frac{a}{b}$
• $\underset{x\to 0}{lim}\frac{{\log }_c(1+ax)}{bx}=\frac{a}{b\ln c}$

L'Hôpital's rule

Main article: l'Hôpital's rule

This rule uses derivatives to find limits of indeterminate forms 0/0 or ±∞/∞, and only applies to such cases. Other indeterminate forms may be manipulated into this form. Given two functions f(x) and g(x), defined over an open interval I containing the desired limit point c, then if:

1. $\underset{x\to c}{lim}f(x)=\underset{x\to c}{lim}g(x)=0,$ or $\underset{x\to c}{lim}f(x)=\pm \underset{x\to c}{lim}g(x)=\pm \mathrm{\infty }$, and
2. $f$ and $g$ are differentiable over $I\setminus \{c\}$, and
3. $g^{\prime }(x)\ne 0$ for all $x\in I\setminus \{c\}$, and
4. $\underset{x\to c}{lim}\frac{f^{\prime }(x)}{g^{\prime }(x)}$ exists,

then:

$\underset{x\to c}{lim}\frac{f(x)}{g(x)}=\underset{x\to c}{lim}\frac{f^{\prime }(x)}{g^{\prime }(x)}$.

Normally, the first condition is the most important one.

For example: $\underset{x\to 0}{lim}\frac{\sin (2x)}{\sin (3x)}=\underset{x\to 0}{lim}\frac{2\cos (2x)}{3\cos (3x)}=\frac{2\cdot 1}{3\cdot 1}=\frac{2}{3}.$

Summations and integrals

Specifying an infinite bound on a summation or integral is a common shorthand for specifying a limit.

A short way to write the limit $\underset{n\to \mathrm{\infty }}{lim}\sum _{i=s}^{n}f(i)$ is $\sum _{i=s}^{\mathrm{\infty }}f(i)$. An important example of limits of sums such as these are series.

A short way to write the limit $\underset{x\to \mathrm{\infty }}{lim}{\int }_a^{x}f(t)dt$ is ${\int }_a^{\mathrm{\infty }}f(t)dt$.

A short way to write the limit $\underset{x\to -\mathrm{\infty }}{lim}{\int }_x^{b}f(t)dt$ is ${\int }_{-\mathrm{\infty }}^{b}f(t)dt$.

User-centered design

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/User-centered_design 

 

See also: Human-centered design

User-centered design (UCD) or user-driven development (UDD) is a framework of process (not restricted to interfaces or technologies) in which usability goals, user characteristics, environment, tasks and workflow of a product, service or process are given extensive attention at each stage of the design process. These tests are conducted with/without actual users during each stage of the process from requirements, pre-production models and post production, completing a circle of proof back to and ensuring that "development proceeds with the user as the center of focus." Such testing is necessary as it is often very difficult for the designers of a product to understand intuitively the first-time users of their design experiences, and what each user's learning curve may look like. User-centered design is based on the understanding of a user, their demands, priorities and experiences and when used, is known to lead to an increased product usefulness and usability as it delivers satisfaction to the user.

The chief difference from other product design philosophies is that user-centered design tries to optimize the product around how users can, want, or need to use the product so that users are not forced to change their behavior and expectations to accommodate the product. The users thus stand in the center of two concentric circles. The inner circle includes the context of the product, objectives of developing it and the environment it would run in. The outer circle involves more granular details of task detail, task organization, and task flow.

History

The term "User-Centered Design" was coined by Rob Kling in 1977 and later adopted in Donald A. Norman's research laboratory at the University of California, San Diego. The concept became widely popular as a result of the publication of the book User-Centered System Design: New Perspectives on Human-Computer Interaction in 1986. The concept gained further attention and acceptance in Norman's seminal book The Design of Everyday Things (originally called The Psychology of Everyday Things). In this book, Norman describes the psychology behind what he deems 'good' and 'bad' design through examples. He exalts the importance of design in our everyday lives, and the consequences of errors caused by bad designs.

The two books include principles for building well-designed products. His recommendations are based on the needs of the user, leaving aside what he considers secondary issues like aesthetics. The main highlights of these are:

1. Simplifying the structure of the tasks such that the possible actions at any moment are intuitive.
2. Make things visible, including the conceptual model of the system, actions, results of actions and feedback.
3. Getting the mappings right between intended results and required actions.
4. Embracing and exploiting the constraints of systems.

In a later book, Emotional Design, Norman returns to some of his earlier ideas to elaborate what he had come to find as overly reductive.

Models and approaches

For example, the User-Centered Design process can help software designers to fulfill the goal of a product engineered for their users. User requirements are considered right from the beginning and included into the whole product cycle. These requirements are noted and refined through investigative methods including: ethnographic study, contextual inquiry, prototype testing, usability testing and other methods. Generative methods may also be used including: card sorting, affinity diagramming and participatory design sessions. In addition, user requirements can be inferred by careful analysis of usable products similar to the product being designed.

User-Centered Design takes inspiration from the following models:

• Cooperative design: involving designers and users on an equal footing. This is the Scandinavian tradition of design of IT artifacts and it has been evolving since 1970. This is also called Co-design.
• Participatory design (PD), a North American term for the same concept, inspired by Cooperative Design, focusing on the participation of users. Since 1990, there has been a bi-annual Participatory Design Conference.
• Contextual design, "customer-centered design" in the actual context, including some ideas from Participatory design

Here are the principles that help in ensuring a design is user-centered:

1. Design is based upon an explicit understanding of users, tasks and environments.
2. Users are involved throughout design and development.
3. Design is driven and refined by user-centered evaluation.
4. Process is iterative.
5. Design addresses the whole user experience.
6. Design team includes multidisciplinary skills and perspectives.

User-Centered Design Process

The goal of the User-Centered design is to make products which have very high usability. This includes how convenient the product is, in terms of its usage, manageability, effectiveness, and how well the product is mapped to the user requirements. Below are the general phases of User-Centered Design process:

1. Specify context of use: Identify who the primary users of the product are, why they will use the product, what are their requirements and under what environment they will use it.
2. Specify requirements: Once the context is specified, it is time to identify the granular requirements of the product. This is an important phase which can further facilitate the designers to create storyboards, and set important goals to make the product successful.
3. Create design solutions and development: Based on product goals and requirements, start an iterative process of product design and development.
4. Evaluate product: Product designers do usability testing to get users' feedback for the product at every stage of User-Centered Design.

In the next steps, the above procedure is repeated to further finish the product. These phases are general approaches and factors like design goals, team and their timeline, and environment in which the product is developed, determine the appropriate phases for a project and their order. You can either follow a waterfall model, agile model or any other software engineering practice.

Purpose

UCD asks questions about users and their tasks and goals, then uses the findings to make decisions about development and design. UCD of a web site, for instance, seeks to answer the following questions:

• Who are the users of the website?
• What are the users' tasks and goals?
• What are the users' experience levels with the website, and the similar websites?
• What functions do the users need from the website?
• What information might the users need, and in what form do they need it?
• How do users think the website should work?
• What are the extreme environments in which website can be accessed in?
• Is the user multitasking?
• Does the interface utilize different input modes, such as touch, speech, gestures or orientation?

Elements

As an example of UCD viewpoints, the essential elements of UCD of a website usually are the considerations of visibility, accessibility, legibility and language.

Visibility

Visibility helps the user construct a mental model of the document. Models help the user predict the effect(s) of their actions while using the document. Important elements (such as those that aid navigation) should be emphatic. Users should be able to tell from a glance what they can and cannot do with the document.

Accessibility

Users should be able to find information quickly and easily throughout the document, regardless of its length. Users should be offered various ways to find information (such as navigational elements, search functions, table of contents, clearly labeled sections, page numbers, color-coding, etc.). Navigational elements should be consistent with the genre of the document. ‘Chunking' is a useful strategy that involves breaking information into small pieces that can be organized into some type of meaningful order or hierarchy. The ability to skim the document allows users to find their piece of information by scanning rather than reading. Bold and italic words are often used to this end.

Legibility

Text should be easy to read: Through analysis of the rhetorical situation, the designer should be able to determine a useful font-family and font style. Ornamental fonts, text in all capital letters, large or small body text can be hard to read and should be avoided. Text-colouring and bolding can be helpful when used in text-heavy scenarios. High figure-ground contrast between text and background increases legibility. Dark text against a light background is most legible.

Language

Depending on the rhetorical situation, certain types of languages are needed. Short sentences are helpful, as are well-written texts used in explanations and similar bulk-text situations. Unless the situation calls for it, jargon or heavily technical terms should not be used. Many writers will choose to use the active voice, verbs (instead of noun strings or nominals), and a simple sentence structure.

Analysis tools

There are a number of tools that are used in the analysis of User-Centered Design, mainly: personas, scenarios, and essential use cases.

Persona

During the UCD process, a Persona representing the user may be created. A persona is a user archetype used to help guide decisions about product features, navigation, interactions, and even visual design. In most cases, personas are synthesized from a series of ethnographic interviews with real people, then captured in 1-2 page descriptions that include behavior patterns, goals, skills, attitudes, and environment, with a few fictional personal details to bring the persona to life.

For each product, or sometimes for each set of tools within a product, there is a small set of personas, one of whom is the primary focus for the design. There is also what is called a secondary persona, where the character is not the main target of the design, but their needs should be met and problems solved if possible. They exist to help account for further possible problems and difficulties that may occur even though the primary persona is satisfied with their solution. There is also an anti-persona, which is the character that the design is specifically not made for.

Personas are useful in the sense that they create a common shared understanding of the user group that the design process is built around. Also, they help to prioritize the design considerations by providing a context of what the user needs and what functions are simply nice to add and have. They can also provide a human face and existence to a diversified and scattered user group, and help in creating some empathy and adding emotions when referring to the users. However, since personas are a generalized perception of the primary stakeholder group from collected data, the characteristics may be too broad and typical, or too much of an "average Joe". Sometimes, personas can have stereotypical properties also, which may hurt the entire design process. Overall, personas can be a useful tool to be used by designers to make informed design decisions around, opposed to referring to a set of data or a wide range of individuals.

Personas can also be modified all through the UCD of a product, based on user testing and changing environment. This is not an ideal way to use Personas but should not be a taboo either particularly when it becomes apparent that variables surrounding a product's development have changed since the design started and current persona/s may not cater well to the changed conditions.

Scenario

A scenario created in the UCD process is a fictional story about the "daily life of" or a sequence of events with the primary stakeholder group as the main character. Typically, a persona that was created earlier is used as the main character of this story. The story should be specific of the events happening that relate to the problems of the primary stakeholder group, and normally the main research questions the design process is built upon. These may turn out to be a simple story about the daily life of an individual, but small details from the events should imply details about the users, and may include emotional or physical characteristics. There can be the best-case scenario, where everything works out best for the main character, the worst-case scenario, where the main character experiences everything going wrong around him or her, and an average-case scenario, which is the typical life of the individual, where nothing really special or really depressing occurs, and the day just moves on.

Scenarios create a social context in which the personas exist, and also create an actual physical world, instead of imagining a character with internal characteristics from gathered data and nothing else; there is more action involved in the persona's existence. A scenario is also more easily understood by people, since it is in the form of a story, and is easier to follow. Yet, like the personas, these scenarios are assumptions made by the researcher and designer, and is also created from a set of organized data. It is important to ensure that scenarios are created as close as possible to real world scenarios. Nevertheless, it can be sometimes difficult to explain and inform how low level tasks occur, for ex- the thought process of a persona before acting.

Use case

In short, a use case describes the interaction between an individual and the rest of the world. Each use case describes an event that may occur for a short period of time in real life, but may consist of intricate details and interactions between the actor and the world. It is represented as a series of simple steps for the character to achieve his or her goal, in the form of a cause and effect scheme. Use cases are normally written in the form of a chart with two columns: first column labelled actor, second column labelled world, and the actions performed by each side written in order in the respective columns. The following is an example of a use case for performing a song on a guitar in front of an audience.

Actor	World
choose music to play
pick up guitar
	display sheet music
perform each note on sheet music using guitar
	convey note to audience using sound
	audience provides feedback to performer
assess performance and adjust as needed based on audience feedback
complete song with required adjustments
	audience applause

The interaction between actor and the world is an act that can be seen in everyday life, and we take them as granted and don't think too much about the small detail that needs to happen in order for an act like performing a piece of music to exist. It is similar to the fact that when speaking our mother tongue, we do not think too much about grammar and how to phrase words; they just come out since we are so used to saying them. The actions between an actor and the world, notably, the primary stakeholder (user) and the world in this case, should be thought about in detail, and hence use cases are created to understand how these tiny interactions occur.

An essential use case is a special kind of use case, also called an abstract use case. Essential use cases describe the essence of the problem, and deals with the nature of the problem itself. While writing essential use cases, no assumptions about unrelated details should be made. In additions, the goals of the subject should be separated from the process and implementation to reach that particular goal. Below is an example of an essential use case with the same goal as the former example.

Actor	World
choose sheet music to perform
gathers necessary resources
	provides access to resources
performs piece sequentially
	convey and interprets performance
	provides feedback
completes performance

Use cases are useful because they help identify useful levels of design work. They allow the designers to see the actual low level processes that are involved in a certain problem, which makes the problem easier to handle, since certain minor steps and details the user makes are exposed. The designers' job should be to take into consideration these small problems in order to arrive at a final solution that works. Another way to say this is that use cases break complicated tasks into smaller bits, where these bits are useful units. Each bit completes a small task, which then builds up to the final bigger task. Like writing code on a computer, it is easier to write the basic smaller parts and make them work first, and then put them together to finish the larger more complicated code, instead of tackling the entire code from the very beginning.

The first solution is less risky because if something goes wrong with the code, it is easier to look for the problem in the smaller bits, since the segment with the problem will be the one that does not work, while in the latter solution, the programmer may have to look through the entire code to search for a single error, which proves time-consuming. The same reasoning goes for writing use cases in UCD. Lastly, use cases convey useful and important tasks where the designer can now see which one are of higher importance than others. Some drawbacks of writing use cases include the fact that each action, by the actor or the world, consist of little detail, and is simply a small action. This may possibly lead to further imagination and different interpretation of action from different designers.

Also, during the process, it is really easy to oversimplify a task, since a small task derived from a larger task may still consist of even smaller tasks which were missed. Picking up a guitar may involve thinking of which guitar to pick up, which pick to use, and think about where the guitar is located first. These tasks may then be divided into smaller tasks, such as first thinking of what colour of guitar fits the place to perform the piece, and other related details. Tasks may be split further down into even tinier tasks, and it is up to the designer to determine what is a suitable place to stop splitting up the tasks. Tasks may not only be oversimplified, they may also be omitted in whole, thus the designer should be aware of all the detail and all the key steps that are involved in an event or action while writing use cases.