Formulas of Limits

01

Algebraic functions

$\displaystyle \Large \lim_{x \,\to\, a} \large \dfrac{x^n-a^n}{x-a} \,=\, n.a^{n-1}$

$\displaystyle \Large \lim_{x \,\to\, 0} \, \large {(1+x)}^{\huge \frac{1}{x}} \,=\, e$

$\displaystyle \Large \lim_{x \,\to\, \infty} \, \large {\Big(1+\dfrac{1}{x}\Big)}^{\displaystyle x} \,=\, e$

03

Exponential functions

$\displaystyle \Large \lim_{x \,\to\, 0} \large \dfrac{e^{\displaystyle \normalsize x}-1}{x} \,=\, 1$

$\displaystyle \Large \lim_{x \,\to\, 0} \large \dfrac{a^{\displaystyle \normalsize x}-1}{x} \,=\, \log_{e}{a}$

03

Algebraic Trigonometric functions

The limit of ratio of $\sin{x}$ to $x$ rule as $x$ approaches zero.

$\displaystyle \Large \lim_{x \,\to\, 0} \large \dfrac{\sin{x}}{x} \,=\, 1$

$\Large \displaystyle \lim_{x \,\to\, 0} \large \dfrac{\sin^{-1} x}{x} \,=\, 1$