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Question

Solve ${lim}_{x \to \frac{π}{2}} [t a n x l n (s i n x)]$

A

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B

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C

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Solution

The correct option is A $0$
${lim}_{x \to \frac{π}{2}} [t a n x l n (s i n x)]$
${lim}_{x \to \frac{π}{2}} [\frac{l n (s i n x)}{c o t x}]$
Applying L'Hopital's rule, we get:
${lim}_{x \to \frac{π}{2}} [\frac{(\frac{c o s x}{s i n x})}{- c o s e c^{2} x}]$
${lim}_{x \to \frac{π}{2}} [- \frac{c o t x}{c o s e c^{2} x}]$
${lim}_{x \to \frac{π}{2}} [- \frac{c o s x}{c o s e c^{3} x}] .$
Direct substitution now, yields: $\frac{0}{1}$ , which equals $0$ .
Therefore,
${lim}_{x \to \frac{π}{2}} [t a n x l n (s i n x)] = 0$

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