$\lim _{x \rightarrow \frac{\pi}{2}} \frac{\cot x-\cos x}{(\pi-2 x)^{3}}$ equals :

JEE Main · 2017 · 02 Apr 2017 · Offline · Limits

lim_{x \to \frac{π}{2}} \frac{c o t x - c o s x}{( π - 2 x ) ^{3}}

equals :

Answer: (C)

Shift with

t = x - \frac{π}{2}

, so

cot x \to - tan t

cos x \to - sin t

and

π - 2 x \to - 2 t

. The limit becomes

t \to 0 lim \frac{sin t - tan t}{- 8 t ^{3}}

. Using

sin t \approx t - \frac{t ^{3}}{6}

and

tan t \approx t + \frac{t ^{3}}{3}

, the numerator is

- \frac{t ^{3}}{2}

, so the ratio tends to

\frac{- t ^{3} /2}{- 8 t ^{3}} = \frac{1}{16}

JEE Main · 2017 · 02 Apr 2017 · Offline · Single correct option

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