Question

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

Answer 1

Answer: 1

$\underset{x\to 0}{\lim }(cosx{)}^{1/x}=k$
$\Rightarrow \underset{x\to 0}{\lim }\frac{1}{x}log(cosx)=logk$
$\Rightarrow \lim x\to 0\frac{1}{x}\underset{x\to 0}{\lim }logcosx=logk$
$\Rightarrow \underset{x\to 0}{\lim }\frac{1}{x}\times 0=lo{g}_{e}k$
$\Rightarrow k=1$.