First slide

Introduction to limits

Question

The value of $lim_{x \to 0} {\{\tan (\frac{π}{4} + x)\}}^{1 / x},$ is

Moderate

A

$e$
B

$e^{2}$
C

$\frac{2}{e}$
D

$\frac{1}{e^{2}}$

Solution

We have,
$\begin{array}{l} lim_{x \to 0} {\{\tan (\frac{π}{4} + x)\}}^{\frac{1}{x}} = lim_{x \to 0} {(\frac{1 + \tan x}{1 - \tan x})}^{1 / x} \\ = lim_{x \to 0} {(1 + \frac{2 \tan x}{1 - \tan x})}^{1 / x} = e^{lim_{x \to 0} \frac{2 \tan x}{1 - \tan x} \cdot \frac{1}{x}} \\ = e^{lim_{x \to 0} \frac{2}{1 - \tan x} \times \frac{\tan x}{x}} = e^{2} \end{array}$

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