Limit Continuity and Differentiability 1 Question 7
8. $\lim _{x \rightarrow \pi / 2} \frac{\cot x-\cos x}{(\pi-2 x)^{3}}$ equals
(2017 Main)
(a) $\frac{1}{24}$
(b) $\frac{1}{16}$
(c) $\frac{1}{8}$
(d) $\frac{1}{4}$
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Answer:
Correct Answer: 8. (b)
Solution:
- Given, $p=\lim _{x \rightarrow 0^{+}}\left(1+\tan ^{2} \sqrt{x}\right)^{\frac{1}{2 x}} \quad\left(1^{\infty}\right.$ form $)$
$$ \begin{aligned} &=e^{\lim _{x \rightarrow 0^{+}} \frac{\tan ^{2} \sqrt{x}}{2 x}}=e^{\frac{1}{2} \lim _{x \rightarrow 0^{+}} \frac{\tan \sqrt{x}}{\sqrt{x}}^{2}}=e^{\frac{1}{2}} \\ & \therefore \quad \log p=\log e^{\frac{1}{2}}=\frac{1}{2} \end{aligned} $$
