MCQ
$\mathop {\lim }\limits_{x \to \infty } \left( {\left| {{x^2}} \right| + x} \right)\log \left( {x{{\cot }^{ - 1}}x} \right)$ =
- A$\frac{1}{3}$
- ✓$ - \frac{1}{3}$
- C$\frac{2}{3}$
- D$ - \frac{2}{3}$
$\mathop {\lim }\limits_{h \to 0} \frac{{h + 1}}{{{h^2}}}.\log \frac{{{{\tan }^{ - 1}}h}}{h}$
$ = \log {\left( {\frac{{{{\tan }^{ - 1}}h}}{h}} \right)^{\frac{{h + 1}}{{{h^2}}}}}$
$ \Rightarrow \log {\left( {1 + \left( {\frac{{{{\tan }^{ - 1}}h}}{h}} \right)} \right)^{\frac{{h + 1}}{{{h^2}}}}}$
$ \Rightarrow \left( {\frac{{{{\tan }^{ - 1}}h}}{h} - 1} \right)\frac{{h + 1}}{{{h^2}}}$
$ = \frac{{\left( {{{\tan }^{ - 1}}h - h} \right)}}{h} \times \frac{{h + 1}}{{{h^2}}}$
Applying series of ${\tan ^{ - 1}}x$
$\mathop {\lim }\limits_{h \to 0} \left[ { - \frac{1}{3} + \frac{{{h^2}}}{5}.....} \right]\left[ {h + 1} \right] = - \frac{1}{3}$
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