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  • If the function   <img alt="\mathrm{f(x)=\left\{\begin{array}{cc}(\cos x)^{1 / x}, x \neq 0 \\ k, x=0\end{array}\right.}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7B

If the function   \mathrm{f(x)=\left\{\begin{array}{cc}(\cos x)^{1 / x}, x \neq 0 \\ k, x=0\end{array}\right.}   is continuous at x=0, then the value of k is

Option: 1

0


Option: 2

1


Option: 3

-1


Option: 4

e


Answers (1)

best_answer

For f(x) to be continuous at x=0, we must have
\mathrm{ \lim _{x \rightarrow 0} f(x)=f(0) }
\mathrm{ \Rightarrow \lim _{x \rightarrow 0}(\cos x)^{1 / x}=k \\ }

\mathrm{ \Rightarrow \lim _{x \rightarrow 0}[1+(\cos x-1)]^{1 / x}=k \Rightarrow e^{\lim _{x \rightarrow 0} \frac{\cos x-1}{x}}=k \\ }
\mathrm{\Rightarrow e^{-\lim _{x \rightarrow 0} \frac{2 \sin ^2 x / 2}{x}}=k }

\mathrm{\Rightarrow e^0=k \Rightarrow k=1 . }
 

Posted by

Rakesh

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