First slide

Introduction to limits

Question

The value of $lim_{x \to 1} (2 - x)^{\tan \frac{π x}{2}}$ is equal to

Moderate

A

$e^{- 2 / π}$
B

$e^{1 / π}$
C

$e^{2 / π}$
D

$e^{- 1 / π}$

Solution

We have,
$\begin{array}{l} lim_{x \to 1} (2 - x)^{\tan \frac{π x}{2}} = lim_{x \to 1} {1 + (1 - x)}^{\tan \frac{π x}{2}} = e^{lim_{x \to 1} (1 - x) \tan (\frac{π}{2} - \frac{π x}{2})} \\ = e^{lim_{h \to 0} h \cot \frac{π h}{2} = e^{lim_{h \to 0} \frac{h}{\tan (\frac{π h}{2})}} = e^{2 / π}} \end{array}$

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