Evaluate

To evaluate:

$\lim _{x \rightarrow 0} \frac{a^{x}-a^{-x}}{x}$

Formula used:

L'Hospital's rule

Let $f(x)$ and $g(x)$ be two functions which are differentiable on an open interval I except at a point a where

$\lim _{x \rightarrow a} \mathrm{f}(\mathrm{x})=\lim _{x \rightarrow a} \mathrm{~g}(\mathrm{x})=0$ or $\pm \infty$

then

$\lim _{x \rightarrow a} \frac{f(x)}{g(x)}=\lim _{x \rightarrow a} \frac{f^{\prime}(x)}{g^{\prime}(x)}$

As $\mathrm{x} \rightarrow 0$, we have

$\lim _{x \rightarrow 0} \frac{a^{x}-a^{-x}}{x}=\frac{0}{0}$

This represents an indeterminate form. Thus applying L'Hospital's rule, we get

$\lim _{x \rightarrow 0} \frac{a^{x}-a^{-x}}{x}=\lim _{x \rightarrow 0} \frac{\frac{d}{d x}\left(a^{x}-a^{-x}\right)}{\frac{d}{d x}(x)}$

$\lim _{x \rightarrow 0} \frac{a^{x}-a^{-x}}{x}=\lim _{x \rightarrow 0} \frac{a^{x} \ln a+a^{-x} \ln a}{1}$

$\lim _{x \rightarrow 0} \frac{a^{x}-a^{-x}}{x}=2 \ln a$

Thus, the value of $\lim _{x \rightarrow 0} \frac{a^{x}-a^{-x}}{x}$ is $2 \ln a$.