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  4. If I= lim limits x arrow 0 sin ((ex-x-1-(x2/2)/x2)), then limit

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Q. If $I=\lim\limits _{x \rightarrow 0} \sin \left(\frac{e^{x}-x-1-\frac{x^{2}}{2}}{x^{2}}\right)$, then limit

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Solution:

$\lim \limits_{x \rightarrow 0} \sin \left(\frac{e^{x}-x-1-\frac{x^{2}}{2}}{x^{2}}\right)$
$=\lim \limits_{x \rightarrow 0} \frac{e^{x}-x-1-\frac{x^{2}}{2}}{x^{2}}$
$=\lim \limits_{x \rightarrow 0} \frac{e^{x}-1-0-x}{2 x}$
$=\lim \limits_{x \rightarrow 0} \frac{e^{x}-1}{2}=0$