limx→π2cotx−cosx(π−2x)3 is equal to

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Question

$lim x \to \frac{π}{2} \frac{cot x - cos x}{(π - 2 x)^{3}}$ is equal to

Options:

A .

1

B .

\frac{π}{2}

C .

\frac{1}{16}

D .

0

Answer: Option C
:
C

$lim x \to \frac{π}{2} \frac{cot x - cos x}{(π - 2 x)^{3}}$
Let $x = \frac{π}{2} + t$
If $x \to \frac{π}{2}, t \to 0$
$lim t \to 0 \frac{sin t - tan t}{- 8 t^{3}}$
$= lim t \to 0 \frac{(t - \frac{t^{3}}{3!} + \frac{t^{5}}{5!} + \dots) - (t + \frac{t^{3}}{3} + \frac{2 t^{5}}{15} + \dots)}{- 8 t^{3}}$
$= \frac{1}{16}$
We can put $x = \frac{π}{2} - t$ and we'll get L.H.L also same.
Since L.H.L = R.H.L the limit exists and is $= \frac{1}{16}$

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