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

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.

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

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

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

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

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

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.

Latest Math Topics

Aug 31, 2024

Aug 07, 2024

Jul 24, 2024

Dec 13, 2023

Latest Math Problems

Sep 04, 2024

Jan 30, 2024

Oct 15, 2023

Copyright © 2012 - 2023 Math Doubts, All Rights Reserved