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Evaluate : $\lim\limits_{x\to 0}\large\frac{e^x-e^{x\cos x}}{x+\sin x}$

$(a)\;1\qquad(b)\;0\qquad(c)\;-1\qquad(d)\;None\;of\;these$

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$\lim\limits_{x\to 0}\large\frac{e^x-1)(e^{x\cos x}-1)}{x+\sin x}$
$\Rightarrow \lim\limits_{x\to 0}\bigg[\large\frac{e^x-1}{x(1+\sin x/x)}-\frac{e^{x\cos x}}{x\cos x(\sec x+\sin x/x\cos x)}\bigg]$
$\lim\limits_{x\to 0}\large\frac{e^x-1}{x}$$=1$
$\Rightarrow \large\frac{1}{1+1}-\frac{1}{1+1}$
$\Rightarrow 0$
Hence (b) is the correct answer.
answered Dec 23, 2013 by sreemathi.v
 

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