Limit of a function

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

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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 which may or may not be in the domain of the function.

Formal definitions, first devised in the early 19th century, are given below. Informally, a function assigns an output to every input . We say that the function has a limit at an input , if gets closer and closer to as moves closer and closer to . More specifically, the output value can be made arbitrarily close to if the input to is taken sufficiently close to . On the other hand, if some inputs very close to 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 Bernard Bolzano who, in 1817, introduced the basics of the epsilon-delta technique (see (ε, δ)-definition of limit below) to define continuous functions. However, his work was not known during his lifetime. Bruce Pourciau argues that Isaac Newton, in his 1687 Principia, demonstrates a more sophisticated understanding of limits than he is generally given credit for, including being the first to present an epsilon argument.

In his 1821 book , Augustin-Louis Cauchy discussed variable quantities, infinitesimals and limits, and defined continuity of <math>y=f(x)</math> by saying that an infinitesimal change in necessarily produces an infinitesimal change in , while Grabiner claims that he used a rigorous epsilon-delta definition in proofs. In 1861, Karl Weierstrass first introduced the epsilon-delta definition of limit in the form it is usually written today. He also introduced the notations <math display="inline">\lim</math> and <math display="inline">\textstyle \lim\limits_{x \to x_0}. \displaystyle</math>

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

Functions of a single variable

Informally, a function <math>f(x)</math> has limit <math>L</math> as <math>x</math> approaches <math>a</math> if <math>f(x)</math> approximates <math>L</math> for <math>x</math> near <math>a</math>. More precisely, the value of <math>f(x)</math> is within a given tolerance of <math>L</math>, provided <math>x</math> is within a corresponding tolerance of <math>a</math>. These two tolerances are often denoted, respectively, by <math>\varepsilon</math> (the tolerance in the value of <math>f(x)</math>) and <math>\delta</math> (the corresponding tolerance in <math>x</math>). The value of the function at <math>x=a</math> is usually omitted from the approximation; for example, in many cases where limits are useful, the function has no value at <math>x=a</math> (it is undefined there).

-definition of limit

thumb|For the depicted , , and , we can ensure that the value is within an arbitrarily small interval by restricting to a sufficiently small interval Hence as .

Suppose <math>f: \R \rightarrow \R</math> is a function defined on the real line, and there are two real numbers and . One would say: "The limit of of , as approaches , exists, and it equals ". and write,

or alternatively, say " tends to as tends to ", and write,

if the following property holds: for every real , there exists a real such that for all real , implies . Let <math>f : S \to \R</math> be a real-valued function defined on some <math>S \subseteq \R.</math> Let be a limit point of some <math>T \subset S</math>—that is, is the limit of some sequence of elements of distinct from . Then we say the limit of , as approaches from values in , is , written

if the following holds:

<math display=block>(\forall \varepsilon > 0 )\, (\exists \delta > 0) \,(\forall x \in T)\, (0 < |x - p| < \delta \implies |f(x) - L| < \varepsilon).</math>

Note, can be any subset of , the domain of . And the limit might depend on the selection of . This generalization includes as special cases limits on an interval, as well as left-handed limits of real-valued functions (e.g., by taking to be an open interval of the form ), and right-handed limits (e.g., by taking to be an open interval of the form ). It also extends the notion of one-sided limits to the included endpoints of (half-)closed intervals, so the square root function <math>f(x) = \sqrt x</math> can have limit 0 as approaches 0 from above:

<math display=block>\lim_{ {x\to 0} \atop {x\in [0, \infty)} } \sqrt{x} = 0</math>

since for every , we may take such that for all , if , then .

This definition allows a limit to be defined at limit points of the domain , if a suitable subset which has the same limit point is chosen.

Notably, the previous two-sided definition works on <math>\operatorname{int} S \cup \operatorname{iso} S^c,</math> which is a subset of the limit points of .

For example, let <math>S = [0,1)\cup (1, 2].</math> The previous two-sided definition would work at <math>1 \in \operatorname{iso} S^c = \{1\},</math> but it wouldn't work at 0 or 2, which are limit points of .

Deleted versus non-deleted limits

The definition of limit given here does not depend on how (or whether) is defined at . Bartle refers to this as a deleted limit, because it excludes the value of at . The corresponding non-deleted limit does depend on the value of at , if is in the domain of . Let <math>f : S \to \R</math> be a real-valued function. The non-deleted limit of , as approaches , is if

<math display=block>(\forall \varepsilon > 0 )\, (\exists \delta > 0) \, (\forall x \in S)\, (|x - p| < \delta \implies |f(x) - L| < \varepsilon).</math>

The definition is the same, except that the neighborhood now includes the point , in contrast to the deleted neighborhood . 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).

Bartle

Examples

Non-existence of one-sided limit(s)

thumb|right|Function without a limit at an [[Classification of discontinuities|essential discontinuity ]]

The function

<math display=block>f(x)=\begin{cases}

\sin\frac{5}{x-1} & \text{ for } x<1 \\

0 & \text{ for } x=1 \\[2pt]

\frac{1}{10x-10}& \text{ for } x>1

\end{cases}</math>

has no limit at (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, see picture), but has a limit at every other -coordinate.

The function

<math display=block>f(x)=\begin{cases}

1 & x \text{ rational } \\

0 & x \text{ irrational }

\end{cases}</math>

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

Non-equality of one-sided limits

The function

<math display=block>f(x)=\begin{cases}

1 & \text{ for } x < 0 \\

2 & \text{ for } x \ge 0

\end{cases}</math>

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

Limits at only one point

The functions

<math display=block>f(x)=\begin{cases}

x & x \text{ rational } \\

0 & x \text{ irrational }

\end{cases}</math>

and

<math display=block>f(x)=\begin{cases}

|x| & x \text{ rational } \\

0 & x \text{ irrational }

\end{cases}</math>

both have a limit at and it equals 0.

Limits at countably many points

The function

<math display=block>f(x)=\begin{cases}

\sin x & x \text{ irrational } \\

1 & x \text{ rational }

\end{cases}</math>

has a limit at any -coordinate of the form <math>\tfrac{\pi}{2} + 2n\pi,</math> where is any integer.

Limits involving infinity

Limits at infinity

thumb|300px|The limit of this function at infinity exists

Let <math>f:S \to \R</math> be a function defined on <math>S \subseteq \R.</math> The limit of as approaches infinity is , denoted

<math display=block> \lim_{x \to \infty}f(x) = L,</math>

means that:

<math display=block>(\forall \varepsilon > 0 )\, (\exists c > 0) \,(\forall x \in S) \,(x > c \implies |f(x) - L| < \varepsilon).</math>

Similarly, the limit of as approaches minus infinity is , denoted

<math display=block> \lim_{x \to -\infty}f(x) = L,</math>

means that:

<math display=block>(\forall \varepsilon > 0)\, (\exists c > 0) \,(\forall x \in S)\, (x < -c \implies |f(x) - L| < \varepsilon).</math>

For example,

<math display=block> \lim_{x \to \infty} \left(-\frac{3\sin x}{x} + 4\right) = 4</math>

because for every , we can take such that for all real , if , then .

Another example is that

<math display=block> \lim_{x \to -\infty}e^{x} = 0</math>

because for every , we can take such that for all real , if , then .

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 <math>f:S \to\mathbb{R}</math> be a function defined on <math>S\subseteq\mathbb{R}.</math> The statement the limit of as approaches is infinity, denoted

<math display=block> \lim_{x \to p} f(x) = \infty, </math>

means that:

<math display=block>(\forall N > 0)\, (\exists \delta > 0)\, (\forall x \in S)\, (0 < | x-p | < \delta \implies f(x) > N) .</math>

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

<math display=block> \lim_{x \to p} f(x) = -\infty, </math>

means that:

<math display=block>(\forall N > 0) \, (\exists \delta > 0) \, (\forall x \in S)\, (0 < | x-p | < \delta \implies f(x) < -N) .</math>

For example,

<math display=block>\lim_{x \to 1} \frac{1}{(x-1)^2} = \infty</math>

because for every , we can take <math display="inline">\delta = \tfrac{1}{\sqrt{N}\delta} = \tfrac{1}{\sqrt N}</math> such that for all real , if , then .

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

<math display=block> \lim_{x \to \infty} f(x) = \infty,</math> or <math> \lim_{x \to p^+}f(x) = -\infty.</math>

For example,

<math display=block>\lim_{x \to 0^+} \ln x = -\infty</math>

because for every , we can take such that for all real , if , then .

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 for some
a neighborhood of ∞ is defined to contain an interval where and
a neighborhood of is defined in the normal way metric space

In this case, is a topological space and any function of the form <math>f : X \to Y</math> with <math>X, Y \subseteq \overline \R</math> 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 and the projectively extended real line is where a neighborhood of ∞ is a set of the form <math>\{x: |x| > c\}.</math> 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, <math>x^{-1}</math> does not possess a central limit (which is normal):

<math display=block>\lim_{x \to 0^{+{1\over x} = +\infty, \quad \lim_{x \to 0^{-{1\over x} = -\infty.</math>

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

<math display=block>\lim_{x \to 0^{+{1\over x} = \lim_{x \to 0^{-{1\over x} = \lim_{x \to 0}{1\over x} = \infty.</math>

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 <math>\lim_{x \to 0^{-{x^{-1 = -\infty,</math> namely, it is convenient for <math>\lim_{x \to -\infty}{x^{-1 = -0</math> to be considered true.

Such zeroes can be seen as an approximation to infinitesimals.

Limits at infinity for rational functions

thumb|300px|Horizontal asymptote about

There are three basic rules for evaluating limits at infinity for a rational function <math>f(x) = \tfrac{p(x)}{q(x)}</math> (where and are polynomials):

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

If the limit at infinity exists, it represents a horizontal asymptote at . 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 represents a distance, the definition of a limit can be extended to functions of more than one variable. In the case of a function <math>f : S \times T \to \R</math> defined on <math>S \times T \subseteq \R^2,</math> we defined the limit as follows: the limit of as approaches is , written

if the following condition holds:

:For every , there exists a such that for all in and in , whenever <math display=inline>0 < \sqrt{(x-p)^2 + (y-q)^2} < \delta,</math> we have ,

or formally:

<math display=block>(\forall \varepsilon > 0)\, (\exists \delta > 0)\, (\forall x \in S) \, (\forall y \in T)\, (0 < \sqrt{(x-p)^2 + (y-q)^2} < \delta \implies |f(x, y) - L| < \varepsilon)).</math>

Here <math display=inline>\sqrt{(x-p)^2 + (y-q)^2}</math> is the Euclidean distance between and . (This can in fact be replaced by any norm , and be extended to any number of variables.)

For example, we may say

because for every , we can take <math display=inline>\delta = \sqrt \varepsilon</math> such that for all real and real , if <math display=inline>0 < \sqrt{(x-0)^2 + (y-0)^2} < \delta,</math> then .

Similar to the case in single variable, the value of at does not matter in this definition of limit.

For such a multivariable limit to exist, this definition requires the value of approaches along every possible path approaching . In the above example, the function

satisfies this condition. This can be seen by considering the polar coordinates

<math display=block>(x,y) = (r\cos\theta, r\sin\theta) \to (0, 0),</math>

which gives

<math display=block>\lim_{r \to 0} f(r \cos \theta, r \sin \theta) = \lim_{r \to 0} \frac{r^4 \cos^4 \theta}{r^2} = \lim_{r \to 0} r^2 \cos^4 \theta.</math>

Here is a function of r which controls the shape of the path along which is approaching . Since is bounded between [−1, 1], by the sandwich theorem, this limit tends to 0.

In contrast, the function

does not have a limit at . Taking the path , we obtain

while taking the path , we obtain

Since the two values do not agree, does not tend to a single value as approaches .

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 <math>f : S \times T \to \R</math> be defined on <math>S \times T \subseteq \R^2,</math> we say the double limit of as approaches and approaches is , written

if the following condition holds:

\end{array}</math>

These rules are also valid for one-sided limits, including when 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.

<math display=block>\begin{array}{rcl}

q + \infty & = & \infty \text{ if } q \neq -\infty \\[8pt]

q \times \infty & = & \begin{cases}

\infty & \text{if } q > 0 \\

-\infty & \text{if } q < 0

\end{cases} \\[6pt]

\displaystyle \frac q \infty & = & 0 \text{ if } q \neq \infty \text{ and } q \neq -\infty \\[6pt]

\infty^q & = & \begin{cases}

0 & \text{if } q < 0 \\

\infty & \text{if } q > 0

\end{cases} \\[4pt]

q^\infty & = & \begin{cases}

0 & \text{if } 0 < q < 1 \\

\infty & \text{if } q > 1

\end{cases} \\[4pt]

q^{-\infty} & = & \begin{cases}

\infty & \text{if } 0 < q < 1 \\

0 & \text{if } q > 1

\end{cases}

\end{array}</math>

(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 and . These indeterminate forms are:

<math display=block>\begin{array}{cc}

\displaystyle \frac{0}{0} & \displaystyle \frac{\pm \infty}{\pm \infty} \\[6pt]

0 \times \pm \infty & \infty + -\infty \\[8pt]

\qquad 0^0 \qquad & \qquad \infty^0 \qquad \\[8pt]

1^{\pm \infty}

\end{array}</math>

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

Limits of compositions of functions

In general, from knowing that <math>\lim_{y \to b} f(y) = c</math> and <math>\lim_{x \to a} g(x) = b,</math> it does not follow that <math>\lim_{x \to a} f(g(x)) = c.</math>

However, this "chain rule" does hold if one of the following additional conditions holds:

(that is, is continuous at ), or
does not take the value near (that is, there exists a such that if then ).

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

<math display=block>f(x) = g(x) = \begin{cases}

0 & \text{if } x\neq 0 \\

1 & \text{if } x=0

\end{cases}</math>

Since the value at is a removable discontinuity,

<math display=block>\lim_{x \to a} f(x) = 0</math> for all .

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

<math display=block>f(f(x))=\begin{cases}

1 & \text{if } x\neq 0 \\

0 & \text{if } x = 0

\end{cases}</math>

and so

<math display=block>\lim_{x \to a} f(f(x)) = 1</math> for all .

Limits of special interest

Rational functions

For a nonnegative integer and constants <math>a_1, a_2, a_3,\ldots, a_n</math> and <math>b_1, b_2, b_3,\ldots, b_n,</math>

<math display=block>\lim_{x \to \infty} \frac{a_1 x^n + a_2 x^{n-1} + a_3 x^{n-2} + \dots + a_n}{b_1 x^n + b_2 x^{n-1} + b_3 x^{n-2} + \dots + b_n} = \frac{a_1}{b_1}</math>

This can be proven by dividing both the numerator and denominator by . 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

<math display=block>\begin{array}{lcl}

\displaystyle \lim_{x \to 0} \frac{\sin x}{x} & = & 1 \\[4pt]

\displaystyle \lim_{x \to 0} \frac{1 - \cos x}{x} & = & 0

\end{array}</math>

Exponential functions

<math display=block>\begin{array}{lcl}

\displaystyle \lim_{x \to 0} (1+x)^{\frac{1}{x & = & \displaystyle \lim_{r \to \infty} \left(1+\frac{1}{r}\right)^r = e \\[4pt]

\displaystyle \lim_{x \to 0} \frac{e^{x}-1}{x} & = & 1 \\[4pt]

\displaystyle \lim_{x \to 0} \frac{e^{ax}-1}{bx} & = & \displaystyle \frac{a}{b} \\[4pt]

\displaystyle \lim_{x \to 0} \frac{c^{ax}-1}{bx} & = & \displaystyle \frac{a}{b}\ln c \\[4pt]

\displaystyle \lim_{x \to 0^+} x^x & = & 1

\end{array}</math>

Logarithmic functions

<math display=block>\begin{array}{lcl}

\displaystyle \lim_{x \to 0} \frac{\ln(1+x)}{x} & = & 1 \\[4pt]

\displaystyle \lim_{x \to 0} \frac{\ln(1+ax)}{bx} & = & \displaystyle \frac{a}{b} \\[4pt]

\displaystyle \lim_{x \to 0} \frac{\log_c(1+ax)}{bx} & = & \displaystyle \frac{a}{b\ln c}

\end{array}</math>

L'Hôpital's rule

This rule uses derivatives to find limits of indeterminate forms or , and only applies to such cases. Other indeterminate forms may be manipulated into this form. Given two functions and , defined over an open interval containing the desired limit point , then if:

<math>\lim_{x \to c}f(x)=\lim_{x \to c}g(x)=0,</math> or <math>\lim_{x \to c}f(x)=\pm\lim_{x \to c}g(x) = \pm\infty,</math> and
<math>f</math> and <math>g</math> are differentiable over <math>I \setminus \{c\},</math> and
<math>g'(x)\neq 0</math> for all <math> x \in I \setminus \{c\},</math> and
<math>\lim_{x\to c}\tfrac{f'(x)}{g'(x)}</math> exists,

then:

Normally, the first condition is the most important one.

For example:

<math>\lim_{x \to 0} \frac{\sin (2x)}{\sin (3x)} =

\lim_{x \to 0} \frac{2 \cos (2x)}{3 \cos (3x)} =

\frac{2 \sdot 1}{3 \sdot 1} =

\frac{2}{3}.</math>

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 <math>\lim_{n \to \infty} \sum_{i=s}^n f(i) </math>

is <math>\sum_{i=s}^\infty f(i).</math> An important example of limits of sums such as these are series.

A short way to write the limit <math>\lim_{x \to \infty} \int_a^x f(t) \; dt </math>

is <math>\int_a^\infty f(t) \; dt.</math>

A short way to write the limit <math>\lim_{x \to -\infty} \int_x^b f(t) \; dt </math>

Limit of a function

History

Functions of a single variable

-definition of limit

Deleted versus non-deleted limits

Examples

Non-existence of one-sided limit(s)

Non-equality of one-sided limits

Limits at only one point

Limits at countably many points

Limits involving infinity

Limits at infinity

Infinite limits

Alternative notation

Limits at infinity for rational functions

Functions of more than one variable

Ordinary limits

Multiple limits

Limits of compositions of functions

Limits of special interest

Rational functions

Trigonometric functions

Exponential functions

Logarithmic functions

L'Hôpital's rule

Summations and integrals

See also

Notes

References

External links