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  4. lim x → 0 ( x.2x-x /1-cosx) is equal to

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Q. $ \lim_ {{x \to 0}} \frac { x.2^x-x} {1-cosx} $ is equal to

Solution:

$\lim_{x\to0} \frac{x-2^{x} -x}{1 -\cos x} = \lim _{x\to 0} \frac{x\left(2^{x} -1\right)}{1-\cos x} $
= $ \lim _{x\to 0} \frac{x^{2}}{1 -\cos x} . \frac{2^{x} -1}{x} $
= $\lim _{x\to 0} \frac{x^{2} }{1-\cos x}.\lim _{x\to 0} \frac{2^{x} -1}{x}$
= $ \lim _{x\to 0} 2\left(\frac{x/2}{\sin x/2}\right)^{2} . \log2 = 2 \log 2$