First slide

Introduction to limits

Question

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

Moderate

A

1
B

$e$
C

$e^{- 1}$
D

none of these

Solution

We have,
$\begin{array}{l} lim_{x \to 0} {(\frac{1 + \tan x}{1 + \sin x})}^{cosec x} \\ = lim_{x \to 0} (1 + \tan x)^{cosec x} (1 + \sin x)^{- cosec x} \\ = e^{lim_{x \to 0} t an x cosec x lim_{x \to 0} \sin x (- cosec x)} \\ = e^{1} \times e^{- 1} = e^{1} \times e^{- 1} = e^{0} = 1 \end{array}$

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