Limit

Given a real or complex function $f(x)$ and some value $c$ , the limit $\lim_{x\to c} f(x)$ of $f(x)$ as $x$ goes to $c$ is defined to be equal to the (real or complex) number $L$ if and only if for every $\epsilon > 0$ there exists a $\delta \in \mathbb R$ such that if $0<|x-c|<\delta$ then $|f(x)-L|< \epsilon$ .

Intuitive Meaning

The formal definition of a limit given above is not necessarily easy to understand. We can instead offer the following informal explanation: a limit is the value to which the function grows close. For example, $\lim_{x\rightarrow 2}x^2=4$ , because whenever $x$ is close to 2, the function $f(x)=x^2$ grows close to 4. In this case, the limit of the function is exactly equal to the value of the function. That is, $\lim_{x\rightarrow c} f(x) = f(c)$ . This is because the function we chose was a continuous function. However, not all functions have this property. For example, consider the function $f(x)$ over the reals defined to be 0 if $x\neq 0$ and 1 if $x=0$ . Although the value of the function at 0 is 1, the limit $\lim_{x\rightarrow 0}f(x)$ is, in fact, zero. Intuitively, this is because no matter how close we get to zero, as long as we never actually reach zero, $f(x)$ will always be close to (in fact equal to) zero. Note that if our definition required only that $|x-c|<\delta$ , the limit of this function would not exist.

Left and Right Hand Limits

Left and right hand limits are the limits taken as a point is approached from the left and from the right, respectively. The left hand limit is denoted as $\lim_{x\to c^{-}} f(x)$ , and the right hand limit is denoted as $\lim_{x\to c^{+}} f(x)$ .

If the left hand and right hand limits at a certain point differ, than the limit does not exist at that point. For example, if we consider the step function (the greatest integer function) $f(x) = \lfloor x \rfloor$ , we have $\lim_{x\to 0^{+}} \lfloor x \rfloor = 0$ , while $\lim_{x\to 0^{-}} \lfloor x \rfloor = -1$ .

Existence of Limits

Limits do not always exist. For example $\lim_{x\rightarrow 0}\frac{1}{x}$ does not exist, since, in fact, there exists no $\epsilon$ for which there exists $\delta$ satisfying the definition's conditions, since $\left|\frac{1}{x}\right|$ grows arbitrarily large as $x$ approaches 0. However, it is possible for $\lim_{x\rightarrow c} f(x)$ not to exist even when $f$ is defined at $c$ . For example, consider the Dirichlet function, $D(x)$ , defined to be 0 when $x$ is irrational, and 1 when $x$ is rational. Here, $\lim_{x\rightarrow c}D(x)$ does not exist for any value of $c$ . Alternatively, limits can exist where a function is not defined, as for the function $f(x)$ defined to be 1, but only for nonzero reals. Here, $\lim_{x\rightarrow 0}f(x)=1$ , since for $x$ arbitrarily close to 0, $f(x)=1$ .

A limit exists if the left and right hand side limits exist, and are equal.

Other Properties

Let $f$ and $g$ be real functions. Then:

$\lim(f+g)(x)=\lim f(x)+\lim g(x)$
$\lim(f-g)(x)=\lim f(x)-\lim g(x)$
$\lim(f\cdot g)(x)=\lim f(x)\cdot\lim g(x)$
$\lim\left(\frac{f}{g}\right)(x)=\frac{\lim f(x)}{\lim g(x)}$ given that $\lim g(x)\ne 0$ .
If a limit exists, it is unique.

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Limit

Contents

Intuitive Meaning

Left and Right Hand Limits

Existence of Limits

Other Properties

See Also