Question

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

• A. $1$
• B. $-1$
• C. $0$
• D. $e$

Answer 1

Answer: (A)

$\displaystyle \lim_{x \to 0} \, (cos \, x)^{1/x}= k $
$ \Rightarrow $$\displaystyle \lim_{x \to 0} \, \frac{1}{x} log (cos \, x) = log \, k$
$ \Rightarrow $$\displaystyle \lim_{x \to 0} \, \frac{log \, (cos \, x)}{x} = log \, k$
Using L-Hospital's rule
$ \Rightarrow $$\displaystyle \lim_{x \to 0} \frac{1}{cos \, x} (-sin \, x) = log \, k$
$ \Rightarrow $$\displaystyle \lim_{x \to 0} - tan \, x = log_e \, k$
$ \Rightarrow $$k = 1$