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

In mathematics, the mathematical constant $e$ represents a napier’s constant. If $x$ is used to denote a variable, then the natural exponential function is written as $e^x$ mathematically.

The quotient of the subtraction of one from the natural exponential function by the variable is a standard form, which often appears while calculating the limits of the functions in which the natural exponential functions are involved. Hence, a standard limit rule with natural exponential function is introduced in calculus and let’s learn it in detail.

The limit of the quotient of the subtraction of one from the natural exponential function $e^{\displaystyle \normalsize x}$ by the variable $x$ as the value of variable $x$ is closer to zero, is written in the following mathematical form.

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

The limit of the quotient of the subtraction of $1$ from the napier’s constant raised to the power of $x$ by the variable $x$ as $x$ tends to zero is equal to one. It can be called the natural exponential limit rule.

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

The limit rule in which the natural exponential function is involved can be written in terms of any variable. Hence, the natural exponential limit rule can be expressed in the following forms.

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

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

$(3).\,\,\,$ $\displaystyle \large \lim_{z \,\to\, 0}{\normalsize \dfrac{e^{\displaystyle \normalsize z}-1}{z}}$ $\,=\,$ $1$

Let us prove the limit of quotient of difference of $e^{\displaystyle x}$ and $1$ by $x$ as $x$ approaches $0$ is equal to $1$.

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