Question

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

• A. $1$
• B. $-1$
• C. $0$
• D. $e$

Answer 1

Answer: (A)

$\underset{x\to 0}{\lim }(cosx{)}^{1/x}=k$
$\Rightarrow$$\underset{x\to 0}{\lim }\frac{1}{x}log(cosx)=logk$
$\Rightarrow$$\underset{x\to 0}{\lim }\frac{log(cosx)}{x}=logk$
Using L-Hospital's rule
$\Rightarrow$$\underset{x\to 0}{\lim }\frac{1}{cosx}(-sinx)=logk$
$\Rightarrow$$\underset{x\to 0}{\lim }-tanx=lo{g}_{e}k$
$\Rightarrow$$k=1$