Question

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

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

Answer 1

Answer: (B)

$1$

For $f\left(x\right)$ to be continuous at $x=0$, we must have
$\underset{x\to 0}{lim}f\left(x\right)=f\left(0\right)$
$\Rightarrow \underset{x\to 0}{lim}{\left(\cos x\right)}^{1/x}=k$
$\Rightarrow \underset{x\to 0}{lim}{(1+(\cos x-1\left)\right)}^{1/x}=k$
$\Rightarrow e^{\underset{x\to 0}{lim}\frac{\cos x-1}{x}}=k$
$\Rightarrow e^{-\underset{x\to 0}{lim}\frac{2{\sin }^2\frac{x}{2}}{x}}=k$

$\Rightarrow e^{-\underset{x\to 0}{lim}\frac{{\sin }^2\frac{x}{2}}{\frac{x}{2}}}=k$
$\Rightarrow e^0=k$
$\Rightarrow k=1$