$\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{a^{\displaystyle \normalsize x}-1}{x}}$ Rule
Formula
$\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{a^{\displaystyle \normalsize x}-1}{x}}$ $\,=\,$ $\log_{e}{a}$
Introduction
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.
$\,\,\,\therefore\,\,\,\,\,\,$ $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{a^{\displaystyle \normalsize x}-1}{x}}$ $\,=\,$ $\ln{a}$
Uses
It is used as a formula to find the limit of a function in which the exponential function is involved.
Other forms
This standard result in limits can be written in several ways in calculus.
$(1).\,\,$ $\displaystyle \large \lim_{h\,\to\,0}{\normalsize \dfrac{a^{\displaystyle \normalsize h}-1}{h}}$ $\,=\,$ $\ln{(a)}$
$(2).\,\,$ $\displaystyle \large \lim_{t\,\to\,0}{\normalsize \dfrac{a^{\displaystyle \normalsize t}-1}{t}}$ $\,=\,$ $\log_{e}{(a)}$
Proof
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.