MCQ
$\mathop {\lim }\limits_{x \to 0} \left[ {\frac{1}{x} - \frac{{\log (1 + x)}}{{{x^2}}}} \right] =$
- ✓$\frac{1}{2}.$
- B$-1/2$
- C$1$
- D$-1$
Aliter : Apply $L-$ Hospital’s rule,
$\mathop {\lim }\limits_{x \to 0} \,\left[ {\frac{{x - \log \,(1 + x)}}{{{x^2}}}} \right]$
$ = \mathop {\lim }\limits_{x \to 0} \frac{{1 - \frac{1}{{1 + x}}}}{{2x}}$
$ = \mathop {\lim }\limits_{x \to 0} \,\,\frac{1}{2}\,{\left( {\frac{1}{{1 + x}}} \right)^2} = \frac{1}{2}.$
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