First slide

Introduction to limits

Question

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

Moderate

A

1
B

$\frac{π}{2}$
C

$\frac{1}{16}$
D

0

Solution

$lim_{x \to \frac{π}{2}} \frac{\cot x - \cos x}{(π - 2 x)^{3}}$
$Let x = t + \frac{π}{2} . If x \to \frac{π}{2} \Rightarrow t \to 0$
$\begin{matrix} lim_{t \to 0} \frac{\sin t - \tan t}{- 8 t^{3}} = lim_{t \to 0} \frac{t - \frac{t^{3}}{1} +,, - t - \frac{t^{3}}{3} \dots}{- 8 t^{3}} \\ = lim_{t \to 0} \frac{- \frac{t^{3}}{2}}{- 8 t^{3}} = \frac{1}{16} \end{matrix}$

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