Evaluate

To evaluate:

$\lim _{x \rightarrow 0} \frac{3^{2+x}-9}{x}$

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

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

As $x \rightarrow 0$, we have

$\lim _{x \rightarrow 0} \frac{3^{2+x}-9}{x}=\frac{0}{0}$

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

$\lim _{x \rightarrow 0} \frac{3^{2+x}-9}{x}=\lim _{x \rightarrow 0} \frac{\frac{d}{d x}\left(3^{2+x}-9\right)}{\frac{d}{d x}(x)}$

$\lim _{x \rightarrow 0} \frac{3^{2+x}-9}{x}=\lim _{x \rightarrow 0} \frac{(2+x) \ln 3}{1}$

$\lim _{x \rightarrow 0} \frac{3^{2+x}-9}{x}=2 \ln 3$

Thus, the value of $\lim _{x \rightarrow 0} \frac{3^{2+x}-9}{x}$ is $2 \ln 3$