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  4. The value of lim limitsx→0 (x cos x - log(1 +x)/x2) is

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Q. The value of $\lim\limits_{x\to0} \frac{x \,cos \,x - log\left(1 +x\right)}{x^{2}} $ is

Limits and Derivatives

Solution:

$\lim\limits_{x\to0} \frac{x \,cos \,x - log\left(1 +x\right)}{x^{2}} $ $(\frac{0}{0}$ Form)
Using L-Hospital's rule,
$\lim\limits_{x\to0} \frac{cos\,x - x\, sin\,x - \frac{1}{x + 1}}{2x} $, $(\frac{0}{0} $ Form)
Again using L-Hospital's rule,
$= \lim\limits_{x\to0} \frac{ -sin \,x - sin\,x - x\,cos\,x + \frac{1}{(x + 1)^2} }{2}$
$ = \frac{1}{2} = 0.5$