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Special limits: the limits worth remembering

The limits that come up again and again in exercises and exams. The limit is on the left, its value on the right.

0/0 limits near zero

Limit	Value
$x \to 0 lim \frac{sin x}{x}$	$1$
$x \to 0 lim \frac{tan x}{x}$	$1$
$x \to 0 lim \frac{1 - cos x}{x ^{2}}$	$\frac{1}{2}$
$x \to 0 lim \frac{e ^{x} - 1}{x}$	$1$
$x \to 0 lim \frac{a ^{x} - 1}{x}$	$ln a$
$x \to 0 lim \frac{ln ( 1 + x )}{x}$	$1$

The limits of e

Limit	Value
$n \to \infty lim (1 + \frac{1}{n})^{n}$	$e$
$x \to \infty lim (1 + \frac{a}{x})^{x}$	$e^{a}$
$x \to 0 lim (1 + x)^{1/ x}$	$e$

Growth rates

Limit	Value
$x \to \infty lim \frac{x ^{n}}{e ^{x}}$	$0$
$x \to \infty lim \frac{ln x}{x ^{a}}, a > 0$	$0$
$x \to 0^{+} lim x ln x$	$0$
$n \to \infty lim n n$	$1$
$n \to \infty lim n a, a > 0$	$1$

Notes

Most limits near zero come from Taylor series: sin x ≈ x, e^x ≈ 1 + x, ln(1 + x) ≈ x. That is why every ratio in the first table tends to 1.

At infinity, an exponential beats any power, and any positive power beats a logarithm. That is what the last table says.