First slide

Continuity

Question

$If f (x) = \{\begin{array}{ll} (\cos x)^{\frac{1}{\sin x}} for x \neq 0 \\ k for x = 0 \end{array} then value of k, so that f is differentiable at x = 0 is$

Moderate

A

0
B

1
C

$\frac{1}{2}$
D

2

Solution

$\underset{x \to 0}{L t} {(\cos x)}^{\frac{1}{\sin x}} = k$
$k = e^{\underset{x \to 0}{L t}} [\frac{1}{\sin x} \log (\cos x)] = e^{\underset{x \to 0}{L t} \frac{- \tan x}{\cos x}}$
$= e^{0} = 1$

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