# a. Let f(x) = frac{e^{x}-e^{-x}}{x} text{Find} lim_{x rightarrow 0}(e^{x}-e^{-x})/x

a. Let $f\left(x\right)=\frac{{e}^{x}-{e}^{-x}}{x}$
$\text{Find}\underset{x\to 0}{lim}\left({e}^{x}-{e}^{-x}\right)/x$
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Aniqa O'Neill

The given function is, $f\left(x\right)=\left({e}^{x}-{e}^{-x}\right)/x$
Note that, the given function is of the form $\frac{0}{0}$.
Now, apply L hopital’s rule and obtain the value of the limit as shown below.
$\underset{x\to 0}{lim}f\left(x\right)=\underset{x\to 0}{lim}\frac{\frac{d}{dx}\left({e}^{x}-{e}^{-x}\right)}{\frac{d}{dx}\left(x\right)}$
$\underset{x\to 0}{lim}f\left(x\right)=\underset{x\to 0}{lim}\frac{{e}^{x}+{e}^{-x}}{1}$
$\underset{x\to 0}{lim}f\left(x\right)=2$
Thus, the value of $\underset{x\to 0}{lim}f\left(x\right)=2$.