Question

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

• A. 1
• B. –1
• C. 0
• D. e

Answer 1

Answer: (A)

1

Answer 2

$\lim_{x \rightarrow 0}(\cos x)^{1/x} = k \Rightarrow \lim_{x \rightarrow 0}\frac{1}{x}\log(\cos x) = \log k$

$\Rightarrow \lim_{x \rightarrow 0}\frac{1}{x}\lim_{x \rightarrow 0}{logcos}x = \log k \Rightarrow \lim_{x \rightarrow 0}\frac{1}{x} \times 0 = \log_{e}k \Rightarrow k = 1$.