Special Limits

We often need the following limits to evaluate other limits that we encounter in calculus. The best way to remember them is perhaps to learn what the corresponding graphs look like.

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#	Limit Expression	Conditions / Notes
1–2	$lim_{x \to + \infty} x^{n} = + \infty$ , \lim_{x\to -\infty} x^n = \begin{cases} -\infty & \text{if } n \text{ is odd} \\ +\infty & \text{if } n \text{ is even} \end{cases}	Integer $n > 0$
3–4	$lim_{x \to + \infty} \sqrt[n]{x} = + \infty$ , \lim_{x\to -\infty} \sqrt[n]{x} = \begin{cases} -\infty & \text{if } n \text{ is odd} \\ \text{undefined} & \text{if } n \text{ is even} \end{cases}	Radical root $n$
5–6	$lim_{x \to 0^{+}} \frac{1}{x^{r}} = + \infty$ , $lim_{x \to \pm \infty} \frac{1}{x^{r}} = 0$	Exponent $r > 0$
7–10	$lim_{x \to + \infty} b^{x} = + \infty$ ( $b > 1$ ), $lim_{x \to - \infty} b^{x} = 0$ ( $b > 1$ )	Exponential base $b$
11–12	$lim_{x \to 0^{+}} \ln x = - \infty$ , $lim_{x \to + \infty} \ln x = + \infty$	Natural logarithm
13–14	$lim_{x \to {\frac{π}{2}}^{-}} \tan x = + \infty$ , $lim_{x \to {\frac{π}{2}}^{+}} \tan x = - \infty$	Tangent vertical asymptote
15–16	$lim_{x \to + \infty} \arctan x = \frac{π}{2}$ , $lim_{x \to - \infty} \arctan x = - \frac{π}{2}$	Inverse tangent horizontal asymptotes
17	$lim_{x \to 0} \frac{\sin x}{x} = 1$	Angle x in radians

List of Special Limits

$lim_{x \to + \infty} x^{n} = + \infty$
\lim_{x\to -\infty} x^n = \begin{cases} -\infty & \text{if } n \text{ is odd} \\ +\infty & \text{if } n \text{ is even} \end{cases}

Graphs of power functions y = x^n — Graphs of power functions $y = x^{n}$ .

$lim_{x \to + \infty} \sqrt[n]{x} = + \infty$
\lim_{x\to -\infty} \sqrt[n]{x} = \begin{cases} -\infty & \text{if } n \text{ is odd} \\ \text{not defined} & \text{if } n \text{ is even} \end{cases}

Graphs of root functions y = n-th root of x — Graphs of root functions $y = \sqrt[n]{x}$ .

$lim_{x \to 0^{+}} \frac{1}{x^{r}} = + \infty$ and $lim_{x \to 0^{-}} \frac{1}{x^{r}} = + \infty or - \infty$ ( $r > 0$ ). (In 5 and 6, the limits as $x \to 0^{-}$ and $x \to - \infty$ are valid provided that $x^{r}$ is defined; that is, when $x^{r}$ is a real number. When $x < 0$ , $x^{r}$ is defined only for rational numbers $r = m / n$ where $n$ is an odd integer. For example, $x^{1 / 3} = \sqrt[3]{x}$ is defined when $x < 0$ , but $x^{1 / 2} = \sqrt{x}$ is not defined for $x < 0$ .)
$lim_{x \to \pm \infty} \frac{1}{x^{r}} = 0$ ( $r > 0$ )

Graphs of reciprocal power functions y = 1/x^r — Graphs of reciprocal power functions $y = \frac{1}{x^{r}}$ .

$lim_{x \to + \infty} b^{x} = + \infty$ ( $b > 1$ )
$lim_{x \to - \infty} b^{x} = 0$ ( $b > 1$ )
$lim_{x \to + \infty} b^{x} = 0$ ( $b < 1$ )
$lim_{x \to - \infty} b^{x} = + \infty$ ( $b < 1$ )

Graphs of exponential functions $y = b^{x}$ .

$lim_{x \to 0^{+}} \ln x = - \infty$
$lim_{x \to + \infty} \ln x = + \infty$

Graph of y = ln(x) — Graph of $y = \ln x$ .

$lim_{x \to {\frac{π}{2}}^{-}} \tan x = + \infty$
$lim_{x \to {\frac{π}{2}}^{+}} \tan x = - \infty$

Graph of y = tan(x) — Graph of $y = \tan x$ .

$lim_{x \to + \infty} \arctan x = \frac{π}{2}$ (Some books denote the inverse of tangent by $\tan^{- 1}$ instead of $\arctan$ .)
$lim_{x \to - \infty} \arctan x = - \frac{π}{2}$

Graph of y = arctan(x) — Graph of $y = \arctan x$ .

$lim_{𝒙 \to 0} \frac{\sin 𝒙}{𝒙} = 1$ ( $x$ is in radians, NOT degrees)

Graph of y = sin(x)/x — Graph of $y = \frac{\sin x}{x}$ .

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List of Special Limits

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