Math Doubts

Power Rule of Limits


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

The limit of power of a function is equal to the power of limit of the function. It is called the power rule of limits.


$x$ is a variable and $f(x)$ is a function in terms of $x$. The literals $n$ and $a$ are constants. The function $f(x)$ and a constant $n$ formed a power function $\Big(f(x)\Big)^{\displaystyle n}$. The limit of the power function $\Big(f(x)\Big)^{\displaystyle n}$ as the input $x$ approaches $a$ is written in mathematics in the following form.

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

The limit of $n$-th power of function $f(x)$ as $x$ tends to $a$ is equal to the $n$-th power of the limit of the function $f(x)$.

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


$(1) \,\,\,\,\,\,$ $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize {(x+6)}^2}$ $\,=\,$ $\Bigg[\displaystyle \large \lim_{x \,\to\, 0} \, {\normalsize {(x+6)}\Bigg]}^2$

$(2) \,\,\,\,\,\,$ $\displaystyle \large \lim_{x \,\to\, 2}{\normalsize {(x^2+3x+4)}^5}$ $\,=\,$ $\Bigg[\displaystyle \large \lim_{x \,\to\, 2} \, {\normalsize {(x^2+3x+4)}\Bigg]}^5$

$(3) \,\,\,\,\,\,$ $\displaystyle \large \lim_{x \,\to\, \infty}{\normalsize {\Bigg(7+\dfrac{2}{x}\Bigg)}^{123}}$ $\,=\,$ $\Bigg[\displaystyle \large \lim_{x \,\to\, \infty} \, {\normalsize {\Bigg(7+\dfrac{2}{x}\Bigg)}\Bigg]}^{123}$


Learn how to derive the power rule of limits in mathematical form in calculus.

Math Doubts

A best free mathematics education website that helps students, teachers and researchers.

Maths Topics

Learn each topic of the mathematics easily with understandable proofs and visual animation graphics.

Maths Problems

A math help place with list of solved problems with answers and worksheets on every concept for your practice.

Learn solutions

Subscribe us

You can get the latest updates from us by following to our official page of Math Doubts in one of your favourite social media sites.

Copyright © 2012 - 2022 Math Doubts, All Rights Reserved