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

Let a variable is denoted by a literal $x$. Now, the natural exponential function is written as the mathematical constant $e$ raised to the power of $x$ in mathematics.

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

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

The limit of this special rational expression with natural exponential function is indeterminate when we try to find the limit by direct substitution.

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

In fact, the limit is not indeterminate but the limit of $e$ raised to the power of $x$ minus $1$ divided by $x$ is equal to one, as the value of $x$ is closer to zero.

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

The limit rule that consists of a natural exponential function is also written popularly in the following forms.

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

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

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

Learn how to derive the limit of the exponential function minus one divided by a variable is equal to one, as the value of variable approaches zero.

Latest Math Topics

Dec 13, 2023

Jul 20, 2023

Jun 26, 2023

Latest Math Problems

Jan 30, 2024

Oct 15, 2023

Copyright © 2012 - 2023 Math Doubts, All Rights Reserved