Math Doubts

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.

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


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

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


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

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


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

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


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

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