Math Doubts

Proof of Limit Rule of an Exponential function

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.

Define the Limit of a function

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.

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

Define the Limit of an Exponential function

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 $a$.

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

Limit Rule of an Exponential function

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.

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