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


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


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.

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

Other forms

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

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

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


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.

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