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Evaluate : $\lim\limits_{x\to 0}(1+\sin x)^{\Large\frac{1}{x^2}}$

$(a)\;0\qquad(b)\;1\qquad(c)\;does\;not\;exist\qquad(d)\;None$

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$\lim\limits_{x\to 0}(1+\sin x)^{\Large\frac{1}{x^2}}=e^{\large\lim\limits_{x\to 0}\big(\large\frac{\sin x}{x^2}\big)}$
$\Rightarrow e^{\large\lim\limits_{x\to 0}\large\frac{1}{x}\big(\frac{\sin x}{x}\big)}$
$\Rightarrow \left\{\begin{array}{1 1}0,&when\;x\to 0^-\\\infty,&when\;x\to 0^+\end{array}\right.$
$\Rightarrow$ Given limit does not exist.
Hence (c) is the correct answer.
answered Dec 23, 2013 by sreemathi.v
 

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