If the function f(x)= ( x)^ 1 x & ; x 0 k & ; x=0 . is continuous… - Vidyadip

MCQ

If the function $f(x)=\left\{\begin{aligned}(\cos x)^{\frac{1}{x}} & ; x \neq 0 \\ k & ; x=0\end{aligned}\right.$ is continuous at $x=0$, then the value of k is

✓
1
B
-1
C
$0$
D
e

✓

Answer

Correct option: A.

1

(A)
Since $f (x)$ is continuous at $x=0$.
$\therefore \quad f(0)=\lim _{x \rightarrow 0} f(x)$
$\Rightarrow k =\lim _{x \rightarrow 0}(\cos x)^{\frac{1}{x}}$
$\Rightarrow \log k =\lim _{x \rightarrow 0} \frac{1}{x} \log (\cos x)$
Applying L'Hospital rule on R.H.S., we get
$\log k =\lim _{x \rightarrow 0} \frac{-\frac{\sin x}{\cos x}}{1}$
$\Rightarrow \log k =0$
$\Rightarrow k = e ^0=1$

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