Limit Formulas

Limit expression	Condition	Result
$\lim_{x \to \infty} e^x$	$x \rightarrow \infty$	$\infty$
$\lim_{x \to -\infty} e^x$	$x \rightarrow -\infty$	0
$\lim_{x \to \infty} \ln(x)$	$x \rightarrow \infty$	$\infty$
$\lim_{x \to 0^{+}} \ln(x)$	$x \rightarrow 0^{+}$	$-\infty$
$\lim_{x \to \infty} \frac{b}{x^r}$	$r > 0$	0
$\lim_{x \to -\infty} \frac{b}{x^r}$	$r > 0$ and $x^r$ is real for negative $x$	0
$\lim_{x \to \infty} x^n$	$n$ even	$\infty$
$\lim_{x \to \infty} x^n$	$n$ odd	$\infty$
$\lim_{x \to -\infty} x^n$	$n$ odd	$-\infty$
$\lim_{x \to \infty} ax^n + \dots + b$	$n$ even	$\text{sgn}(a)\infty$
$\lim_{x \to \infty} ax^n + \dots + b$	$n$ odd	$\text{sgn}(a)\infty$
$\lim_{x \to -\infty} ax^n + \dots + b$	$n$ odd	$-\text{sgn}(a)\infty$

Note: $\text{sgn}(a)$ is the sign function, which is 1 if $a > 0$ and -1 if $a < 0$.

References

• calculus_cheat_sheet_limits | tutorial.math.lamar.edu