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  • If  <img alt="f(x)=\left\{\begin{array}{lll} (\cos x)^{1 / \sin x} \text { for } x \neq 0 \\ k \,\,\,\,\,\,\,\,\,

If  f(x)=\left\{\begin{array}{lll} (\cos x)^{1 / \sin x} \text { for } x \neq 0 \\ k \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text { for } x=0 \end{array}\right.

The value of k, so that f is continuous at x=0 is

Option: 1

0


Option: 2

1


Option: 3

1/2


Option: 4

None of these


Answers (1)

best_answer

Given f(0)=k

\lim _{x \rightarrow 0} f(x)=\lim _{x \rightarrow 0}(\cos x)^{1 / \sin x} \quad\,\,\,\, \,{ [1^{\infty} form] } \\

=\lim _{x \rightarrow 0} e^{\frac{1}{\sin x}(\cos x-1)} \\

 =\lim _{x \rightarrow 0}(e)^{\left(-2 \sin ^{2} x / 2\right) /((2 \sin x / 2) \cos x / 2)} \\

 =\lim _{x \rightarrow 0} e^{-\tan \frac{x}{2}}=e^{0}=1

For f(x) to be continuous at x=0, k=1.

Hence (b) is the correct answer.

Posted by

Ritika Kankaria

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