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  4. lim x → 0 (x cos x-log (1+x)/x2) is equal to

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Q. $ \lim_ {{x \to 0}} \frac {x\,cos\, x-log (1+x)}{x^2} $ is equal to

Solution:

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