1. Tardigrade
  2. Question
  3. Mathematics
  4. If the function f(x) = begincases (cos x)1/x x ≠ 0 [2ex] k, x = 0 endcases is continuous at x = 0 then the value of k is

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Q. If the function $f(x) = \begin{cases} (cos\,x)^{1/x} & x \ne 0 \\[2ex] k, & x = 0 \end{cases}$ is continuous at $ x = 0$ then the value of $k$ is

Continuity and Differentiability

Solution:

$\lim\limits_{x \to 0}(cos\,x)^{1/x} = k$
$\Rightarrow \lim\limits_{x \to 0} \frac{1}{x} log (cos\,x) = log\,k$
$\Rightarrow \lim\limits{x \to 0} \frac{1}{x} \lim\limits_{x\to 0} log \,cos\,x = log\,k$
$\Rightarrow \lim\limits_{x\to 0} \frac{1}{x} \times 0 = log_e \,k $
$\Rightarrow k = 1$.