The literals $a$ and $b$ are two constants, and $x$ is a variable. A function in terms of $x$ is denoted by $f(x)$ in mathematics. An exponential function in terms of constant $b$ and function $f(x)$ is written as $b^{\displaystyle f(x)}$ mathematically. In this case, $a$ is a value of the variable $x$.

Now, let’s start deriving the limit rule of an exponential function in calculus mathematically.

The limit of a function $f(x)$ as the input $x$ approaches $a$ is written as follows.

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

Take, the limit of the function as $x$ tends to $a$ is equal to $L$.

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

Now, find the limit of the function as $x$ closer to $a$ by direct substitution method.

$\implies$ $L \,=\, f(a)$

Now, write the limit of an exponential function as $x$ approaches $a$ in mathematical form.

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

Use the direct substitution method and evaluate the limit of the exponential function as $x$ approaches $a$.

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

According to the first step, $L = f(a)$. So, replace the value of $f(a)$ by $L$ in the above mathematical equation.

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

Actually, it is taken that $L \,=\, \displaystyle \large \lim_{x \,\to\, a}{\normalsize f(x)}$ in the first step.

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

Therefore, it is proved that the limit of an exponential function is equal to the limit of the exponent with same base. Thus, the limit rule of an exponential function is proved mathematically in calculus.

Latest Math Topics

Jan 06, 2023

Jan 03, 2023

Jan 01, 2023

Dec 26, 2022

Dec 08, 2022

Latest Math Problems

Jan 31, 2023

Nov 25, 2022

Nov 02, 2022

Oct 26, 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