MCQ
$\mathop {\lim }\limits_{x \to 0} \frac{{x{e^x} - \log (1 + x)}}{{{x^2}}} =$
- A$\frac{2}{3}$
- B$\frac{1}{3}$
- C$\frac{1}{2}$
- ✓$\frac{3}{2}$
Applying $ L-$ Hospital's rule,
$y = \mathop {\lim }\limits_{x \to 0} \,\,\frac{{{e^x} + x\,{e^x} - \frac{1}{{1 + x}}}}{{2x}}$, $\left( {\frac{0}{0}\,{\rm{form}}} \right)$
$y = \mathop {\lim }\limits_{x \to 0} \,\,\frac{1}{2}\,\left[ {{e^x} + {e^x} + x\,{e^x} + \frac{1}{{{{(1 + x)}^2}}}} \right]$
$y = \mathop {\lim }\limits_{x \to 0} \,\,\frac{1}{2}\,[1 + 1 + 0 + 1] = \frac{3}{2}$.
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