Basic Maths Grand Test- Q19

0 votes

{\lim}_{x \to \pi/2} \frac{cotx-cosx}{(\pi-2x)^3}$ equals

A. 1/4

B. 1/24

C. 1/16

D. 1/8

asked Nov 18 in Basic Maths by gbmentor (85,520 points)
reshown Nov 20 by getgatebook

1 Answer

+1 vote

\lim_{x\rightarrow \frac{\pi}{2}} \frac{cotx-cosx}{(\pi-2x)^3}

Let\ x =\frac{\pi}{2}-h

= \lim_{h\rightarrow 0} \frac{cot( \frac{\pi}{2}-h)-cos( \frac{\pi}{2}-h)}{(\pi-2( \frac{\pi}{2}-h))^3}

= \lim_{h\rightarrow 0} \frac{tanh-sinh}{8h^3}

= \lim_{h\rightarrow 0} \frac{tanh(1-cosh)}{8h^3}

= \lim_{h\rightarrow 0} \frac{tanh(2sin^2 \frac{h}{2})}{8h^3}

= \lim_{h\rightarrow 0} \frac{tanh}{h}*\frac{2sin^2 \frac{h}{2}}{8h^2}

= \lim_{h\rightarrow 0} \frac{2sin^2 \frac{h}{2}}{8(\frac{h}{2})^2*4}

= \lim_{h\rightarrow 0} \frac{2}{32}*(\frac{sin \frac{h}{2}}{\frac{h}{2}})^2

= \frac {2}{32}

= \frac {1}{16}

So option is correct.

answered Nov 23 by tspranaykumar562 (2,720 points)
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