# Limits rules

The list of fundamental properties and standard results of limits.

## Operations

There are four fundamental limit operations in calculus. Let’s learn each limit operation with proof and examples to understand how to use them in mathematics.

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

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

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

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

Likewise, there are two other limit operations in mathematics. So, let us learn them with proofs and example to learn how to use them mathematically in calculus.

$(5).\,\,$ $\displaystyle \large \lim_{x\,\to\,a}{\normalsize \dfrac{1}{f(x)}}$ $\,=\,$ $\dfrac{1}{\displaystyle \large \lim_{x\,\to\,a}{\normalsize f(x)}}$

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

### Properties

List of fundamental properties of limits in mathematical form.

$(7) \,\,\,\,\,\,$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize e^{f{(x)}}}$ $\,=\,$ $e^{\, \displaystyle \large \lim_{x \,\to\, a}{\normalsize f{(x)}}}$

$(8) \,\,\,\,\,\,$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize {[f{(x)}]}^n}$ $\,=\,$ ${\Big[\displaystyle \large \lim_{x \,\to\, a}{\normalsize f{(x)}}\Big]}^n$

$(9) \,\,\,\,\,\,$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize \sqrt[\displaystyle n]{f{(x)}} }$ $\,=\,$ $\sqrt[\displaystyle n]{ \displaystyle \large \lim_{x \,\to\, a}{\normalsize f{(x)} }}$

$(10) \,\,\,\,\,\,$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize f{(g{(x)})}}$ $\,=\,$ $f{\Big(\displaystyle \large \lim_{x \,\to\, a}{\normalsize g{(x)}}\Big)}$

### Formulas

List of standard results in all branches of mathematics.

#### Exponential Limit Rules

Learn list of limit rules for exponential functions.

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

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

$(3) \,\,\,\,\,\,$ $\displaystyle \large \lim_{x \,\to\, a} \dfrac{x^n-a^n}{x-a}$ $\,=\,$ $n.a^{n-1}$

$(4) \,\,\,\,\,\,$ $\displaystyle \large \lim_{x \,\to\, 0} {(1+x)}^\frac{1}{x}$ $\,=\,$ $e$

$(5) \,\,\,\,\,\,$ $\displaystyle \large \lim_{x \,\to\, \infty} {\Big(1+\dfrac{1}{x}\Big)}^{\displaystyle x}$ $\,=\,$ $e$

#### 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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