Math Doubts

$\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{a^{\displaystyle \normalsize x}-1}{x}}$ Rule

Formula

$\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{a^{\displaystyle \normalsize x}-1}{x}}$ $\,=\,$ $\log_{e}{a}$

Introduction

Let us denote a constant by a literal $a$ and represent a variable by a literal $x$. The exponential function is written as the literal $a$ raised to the power of $x$ in mathematics.

The quotient of the exponential function in $x$ minus one divided by $x$ forms a special function in rational form and its limit is mathematically written as follows, when the value of variable $x$ approaches zero.

$\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{a^{\displaystyle \normalsize x}-1}{x}}$

According to the direct substitution, the limit of $a$ raised to the power of $x$ minus $1$ divided by $x$ is indeterminate, as the value of $x$ tends to $0$.

$\implies$ $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{a^{\displaystyle \normalsize x}-1}{x}}$ $\,=\,$ $\dfrac{0}{0}$

However, the limit of the rational function in which the exponential function is involved, is not indeterminate, as the value of $x$ approaches zero, and the limit is equal to the natural logarithm of constant $a$.

$\implies$ $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{a^{\displaystyle \normalsize x}-1}{x}}$ $\,=\,$ $\log_{e}{a}$

According to the logarithms, the natural logarithm of a can also be written simply as follows.

$\,\,\,\therefore\,\,\,\,\,\,$ $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{a^{\displaystyle \normalsize x}-1}{x}}$ $\,=\,$ $\ln{a}$

Uses

It is used as a formula to find the limit of a function in which the exponential function is involved.

Other forms

This standard result in limits can be written in several ways in calculus.

$(1).\,\,$ $\displaystyle \large \lim_{h\,\to\,0}{\normalsize \dfrac{a^{\displaystyle \normalsize h}-1}{h}}$ $\,=\,$ $\ln{(a)}$

$(2).\,\,$ $\displaystyle \large \lim_{t\,\to\,0}{\normalsize \dfrac{a^{\displaystyle \normalsize t}-1}{t}}$ $\,=\,$ $\log_{e}{(a)}$

Proof

Learn how to prove the limit of a constant raised to the power of a variable minus one divided by a variable as the variable tends to zero is equal to the natural logarithm of the constant.

Math Questions

The math problems with solutions to learn how to solve a problem.

Learn solutions

Math Worksheets

The math worksheets with answers for your practice with examples.

Practice now

Math Videos

The math videos tutorials with visual graphics to learn every concept.

Watch now

Subscribe us

Get the latest math updates from the Math Doubts by subscribing us.

Learn more

Math Doubts

A free math education service for students to learn every math concept easily, for teachers to teach mathematics understandably and for mathematicians to share their maths researching projects.

Copyright © 2012 - 2023 Math Doubts, All Rights Reserved