# Limit rules

## Properties

List of fundamental properties of limits in mathematical form.

$(1) \,\,\,\,\,\,$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize \Big[f{(x)}+g{(x)}\Big]}$ $\,=\,$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize f{(x)}}$ $+$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize g{(x)}}$

$(2) \,\,\,\,\,\,$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize \Big[f{(x)}-g{(x)}\Big]}$ $\,=\,$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize f{(x)}}$ $-$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize g{(x)}}$

$(3) \,\,\,\,\,\,$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize \Big[f{(x)}.g{(x)}\Big]}$ $\,=\,$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize f{(x)}}$ $\times$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize g{(x)}}$

$(4) \,\,\,\,\,\,$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize \dfrac{f{(x)}}{g{(x)}}}$ $\,=\,$ $\dfrac{\displaystyle \large \lim_{x \,\to\, a}{\normalsize f{(x)}}}{\displaystyle \large \lim_{x \,\to\, a}{\normalsize g{(x)}}}$

$(5) \,\,\,\,\,\,$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize {f{(x)}}^{g{(x)}}}$ $\,=\,$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize {f{(x)}}^{\, \displaystyle \large \lim_{x \,\to\, a}{\normalsize g{(x)}}}}$

$(6) \,\,\,\,\,\,$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize f{(g{(x)})}}$ $\,=\,$ $f{\Big(\displaystyle \large \lim_{x \,\to\, a}{\normalsize g{(x)}}\Big)}$

### Formulas

List of standard results in all branches of mathematics.

#### Exponential Limit Rules

Learn list of limit rules for exponential functions.

$(1) \,\,\,\,\,\,$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize {f{(x)}}^{g{(x)}}}$ $\,=\,$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize {f{(x)}}^{\, \displaystyle \large \lim_{x \,\to\, a}{\normalsize g{(x)}}}}$

$(2) \,\,\,\,\,\,$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize e^{f{(x)}}}$ $\,=\,$ $e^{\, \displaystyle \large \lim_{x \,\to\, a}{\normalsize f{(x)}}}$

$(3) \,\,\,\,\,\,$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize {[f{(x)}]}^n}$ $\,=\,$ ${\Big[\displaystyle \large \lim_{x \,\to\, a}{\normalsize f{(x)}}\Big]}^n$

$(4) \,\,\,\,\,\,$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize \sqrt[\displaystyle n]{f{(x)}} }$ $\,=\,$ $\sqrt[\displaystyle n]{ \displaystyle \large \lim_{x \,\to\, a}{\normalsize f{(x)} }}$

$(5) \,\,\,\,\,\,$ $\displaystyle \large \lim_{x \,\to\, 0} \normalsize \dfrac{e^{\displaystyle \normalsize x}-1}{x} \,=\, 1$

$(6) \,\,\,\,\,\,$ $\displaystyle \large \lim_{x \,\to\, 0} \normalsize \dfrac{a^{\displaystyle \normalsize x}-1}{x} \,=\, \log_{e}{a}$

$(7) \,\,\,\,\,\,$ $\displaystyle \large \lim_{x \,\to\, a} \dfrac{x^n-a^n}{x-a}$ $\,=\,$ $n.a^{n-1}$

$(8) \,\,\,\,\,\,$ $\displaystyle \large \lim_{x \,\to\, 0} {(1+x)}^\frac{1}{x}$ $\,=\,$ $e$

$(9) \,\,\,\,\,\,$ $\displaystyle \large \lim_{x \,\to\, \infty} {\Bigg(1+\dfrac{1}{x}\Bigg)}^x$ $\,=\,$ $e$

#### Trigonometric Limit Rules

Learn list of limits of trigonometric functions in calculus.

$(1) \,\,\,\,\,\,$ $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{\sin{x}}{x}} \,=\, 1$

$(2) \,\,\,\,\,\,$ $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{\tan{x}}{x}} \,=\, 1$

#### Inverse Trigonometric Limit Rules

Learn list of inverse trigonometric limit rules, used in calculus.

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

$(2) \,\,\,\,\,\,$ $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{\tan^{-1}{x}}{x}} \,=\, 1$

#### Logarithmic Limit Rule

$(1) \,\,\,\,\,\,$ $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{\log_{e}{(1+x)}}{x}} \,=\, 1$

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