Q83 / 170 · Manipal (MET) 2016 · Mathematics

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

  • A
    $8$
  • Correct answer
    B
    $1$
  • C
    $-1$
  • D
    None of these

Solution

For $f(x)$ to be continuous at $x=0$, we must have $\lim _{x \rightarrow 0} f(x)=f(0)$ $\Rightarrow \quad \lim _{x \rightarrow 0}(\cos x)^{1 / x}=k$ $\Rightarrow \quad \lim _{x \rightarrow 0}\{1+(\cos x-1)\}^{1 / x}=k$ $\Rightarrow \quad e^{\lim _{x \rightarrow 0} \frac{\cos x-1}{x}}=k$ $\Rightarrow \quad e^{-\lim _{x \rightarrow 0} \frac{2 \sin ^2 x / 2}{x}}=k$ $\begin{array}{ll}\Rightarrow & e^0=k \\ \therefore & k=1\end{array}$

Hence, the correct answer is option: 'B'.

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