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

## Formula

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

### Introduction

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.

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

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

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

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.

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

#### Other forms

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

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

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

#### Uses

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

##### Proof

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.

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