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  4. displaystyle lim x arrow 0 (ex+e-x+2 cos x-4/x4) is equal to

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Q. $\displaystyle\lim _{x \rightarrow 0} \frac{e^{x}+e^{-x}+2 \cos x-4}{x^{4}}$ is equal to

Limits and Derivatives

Solution:

$\displaystyle\lim _{x \rightarrow 0} \frac{e^{x}+e^{-x}+2 \cos x-4}{x^{4}}$ $\left(\frac{0}{0}\right.$ form $)$
$=\displaystyle\lim _{x \rightarrow 0} \frac{e^{x}-e^{-x}-2 \sin x}{4 x^{3}}$ $\left(\frac{0}{0}\right.$ form $)$
$=\displaystyle\lim _{x \rightarrow 0} \frac{e^{x}+e^{-x}-2 \cos x}{12 x^{2}}$ $\left(\frac{0}{0}\right.$ form $)$
$=\displaystyle\lim _{x \rightarrow 0} \frac{e^{x}-e^{-x}+2 \sin x}{24 x}$ $\left(\frac{0}{0}\right.$ form $)$
$=\displaystyle\lim _{x \rightarrow 0} \frac{e^{x}+e^{-x}+2 \cos x}{24}$ $=\frac{4}{24}=\frac{1}{6}$