Math Doubts

Constant Power rule of Limits

Formula

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

The limit of a function raised to power of a constant is equal to the limit raised to the power of constant. It is called as constant power rule or constant exponent power rule of limits.

Introduction

$f{(x)}$ is a function in terms of $x$ and $n$ is a constant. An exponential function is formed mathematically. In this exponential function, the function $f{(x)}$ is base and the constant $n$ is exponent of the exponential function.

In this special case, the limit of the function $f{(x)}$ is raised to the power of $n$ is equal to the limit raised to the power of $n$. This fundamental property is used as a formula in limits.

Examples

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

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

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

Other form

The property of constant power rule of limits can be written in terms of any function and any constant mathematically and you have to understand that the limit of any function raised to power of any constant is always is equal to the limit raised to the power of that constant.

$(1) \,\,\,\,\,\,$ $\displaystyle \large \lim_{y \,\to\, m}{\normalsize {g{(y)}}^c}$ $\,=\,$ $\Bigg[\displaystyle \large \lim_{y \,\to\, m} \, {\normalsize {g{(y)}}\Bigg]}^c$

$(2) \,\,\,\,\,\,$ $\displaystyle \large \lim_{m \,\to\, c}{\normalsize {j{(m)}}^k}$ $\,=\,$ $\Bigg[\displaystyle \large \lim_{m \,\to\, c} \, {\normalsize {j{(m)}}\Bigg]}^k$

$(3) \,\,\,\,\,\,$ $\displaystyle \large \lim_{h \,\to\, e}{\normalsize {l{(h)}}^q}$ $\,=\,$ $\Bigg[\displaystyle \large \lim_{h \,\to\, e} \, {\normalsize {l{(h)}}\Bigg]}^q$

Proof

Learn how to prove the limit of a function raised to the power of a constant is equal to the limit raised to the power of the constant.



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