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

It is a property of power rule, used to find the limit of an exponential function whose base and exponent are in a function form.

$x$ is a variable and two functions $f{(x)}$ and $g{(x)}$ are defined in terms of $x$. The limits of $f{(x)}$ and $g{(x)}$ as $x$ closer to $a$ are written mathematically in calculus as follows.

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

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

Assume, the functions $f{(x)}$ and $g{(x)}$ are formed a function in exponential form.

${f{(x)}}^{g{(x)}}$

Now, find the limit of this exponential function as $x$ approaches $a$.

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

Find the limit of the exponential function by substituting $x$ by $a$.

$= \,\,\, {f{(a)}}^{g{(a)}}$

The limits of functions $f{(x)}$ and $g{(x)}$ as $x$ tends to $a$ are $f{(a)}$ and $g{(a)}$ respectively. Therefore, it can be written that $f{(a)}$ and $g{(a)}$ as the limits of functions $f{(x)}$ and $g{(x)}$ respectively.

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

Actually, the value of $f{(a)}$ is raised to the power of $g{(a)}$ is determined as the limit of the $f{(x)}$ is raised to the power of $g{(x)}$ as $x$ closer to $a$.

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

Therefore, the limit property is proved that the limit of $f{(x)}$ is raised to the power of $g{(x)}$ as $x$ approaches $a$ equals to the limit of $f{(x)}$ as $x$ approaches $a$ is raised to the power of the limit of $g{(x)}$ as $x$ closer to $a$.

The limit rule is completely in exponential notation. So, it is called as the power rule of limit in calculus.

Latest Math Topics

Jul 24, 2022

Jul 15, 2022

Latest Math Problems

Sep 30, 2022

Jul 29, 2022

Jul 17, 2022

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

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

Learn how to solve the maths problems in different methods with understandable steps.

Copyright © 2012 - 2022 Math Doubts, All Rights Reserved