Special Limits

Limit	Value	Condition
$\underset{x\to +\mathrm{\infty }}{lim}{x}^n$	$+\mathrm{\infty }$	—
$\underset{x\to 0^{+}}{lim}{\displaystyle \frac{1}{x^r}}$	$+\mathrm{\infty }$
$\underset{x\to \pm \mathrm{\infty }}{lim}{\displaystyle \frac{1}{x^r}}$	$0$
$\underset{x\to +\mathrm{\infty }}{lim}{b}^x$	$+\mathrm{\infty }$
$\underset{x\to -\mathrm{\infty }}{lim}{b}^x$	$0$
$\underset{x\to 0}{lim}{\displaystyle \frac{\sin x}{x}}$	$1$	$x$ in radians
$\underset{x\to +\mathrm{\infty }}{lim}\arctan x$	${\displaystyle \frac{\pi }{2}}$	—

Power Functions

1. ${\displaystyle \underset{x\to +\mathrm{\infty }}{lim}{x}^n=+\mathrm{\infty }}$
2. \displaystyle\lim_{x \to -\infty} x^n = \begin{cases} -\infty & \text{if } n \text{ is odd} \\ +\infty & \text{if } n \text{ is even} \end{cases}

Root Functions

3. ${\displaystyle \underset{x\to +\mathrm{\infty }}{lim}\sqrt[n]{x}=+\mathrm{\infty }}$
4. \displaystyle\lim_{x \to -\infty} \sqrt[n]{x} = \begin{cases} -\infty & \text{ifnn$\text{is even} end{cases}}$

Reciprocal Powers

5. ${\displaystyle \underset{x\to 0^{+}}{lim}\frac{1}{x^r}=+\mathrm{\infty }}$ and ${\displaystyle \underset{x\to 0^{-}}{lim}\frac{1}{x^r}=+\mathrm{\infty }\text{ or }-\mathrm{\infty }}$    (, provided $x^r$ is defined for negative $x$)
6. ${\displaystyle \underset{x\to \pm \mathrm{\infty }}{lim}\frac{1}{x^r}=0}$    ()

For limits (5) and (6): when , $x^r$ 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 for , but $x^{1/2}=\sqrt{x}$ is not.

Exponential Functions

7. ${\displaystyle \underset{x\to +\mathrm{\infty }}{lim}{b}^x=+\mathrm{\infty }}$    ()
8. ${\displaystyle \underset{x\to -\mathrm{\infty }}{lim}{b}^x=0}$    ()
9. ${\displaystyle \underset{x\to +\mathrm{\infty }}{lim}{b}^x=0}$    ()
10. ${\displaystyle \underset{x\to -\mathrm{\infty }}{lim}{b}^x=+\mathrm{\infty }}$    ()

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Natural Logarithm

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

Tangent Function

13. ${\displaystyle \underset{x\to \pm {\frac{\pi }{2}}^{-}}{lim}\tan x=+\mathrm{\infty }}$
14. ${\displaystyle \underset{x\to \pm {\frac{\pi }{2}}^{+}}{lim}\tan x=-\mathrm{\infty }}$

Arctangent Function

15. ${\displaystyle \underset{x\to +\mathrm{\infty }}{lim}\arctan x=\frac{\pi }{2}}$    (some books write ${\tan }^{-1}$ instead of $\arctan$)
16. ${\displaystyle \underset{x\to -\mathrm{\infty }}{lim}\arctan x=-\frac{\pi }{2}}$

The Sinc Limit

17. ${\displaystyle \underset{𝒙\mathbf{\to }\mathbf{0}}{\mathbf{lim}}\frac{\mathbf{sin}\mathbf{}𝒙}{𝒙}\mathbf{=}\mathbf{1}}$    ($x$ is in radians, not degrees)

Frequently Asked Questions

Why does $\underset{x\to 0}{lim}\sin x/x=1$ only hold in radians?

The geometric proof (area comparison of a circular sector) uses the arc-length formula $s=r\theta$, which is valid only when $\theta$ is measured in radians. In degrees, the corresponding limit would be $\pi /180$, not $1$.

How do I remember limits 7–10 for exponential functions?

Think about the graph of $b^x$. For the graph rises steeply to the right (so $\underset{x\to +\mathrm{\infty }}{lim}{b}^x=+\mathrm{\infty }$) and flattens to zero on the left (so $\underset{x\to -\mathrm{\infty }}{lim}{b}^x=0$). For the graph is a reflection: it falls to zero on the right and rises steeply on the left.

What are the limits of hyperbolic functions?

The limits of hyperbolic functions ($\sinh x$, $\cosh x$, $\tanh x$, etc.) are covered in the section on hyperbolic functions in the differentiation chapter.