# Limits operations

The fundamental operations are also involved in limits. So, it is essential to learn the basic mathematical operations with their formulas for studying the limits clearly. Here is the list of fundamental operations of limits with their formulas.

## Basic operations

The limit of sum of two or more functions is equal to sum of their limits.

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

#### Subtraction

The limit of difference of any two functions is equal to difference of their limits.

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

#### Multiplication

The limit of product of two or more functions is equal to product of their limits.

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

#### Division

The limit of quotient of two functions is equal to quotient of their limits.

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

### Other operations

#### Reciprocal rule

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

#### Exponentiation

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

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