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STD 12 - 5. continuity and differentiation — JEE STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceJEESTD 12 - 5. continuity and differentiation4 Marks
Question
If the function $f(x) = \left\{ \begin{array}{l}{(\cos x)^{1/x}},\;x \ne 0\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,k,\,x = 0\end{array} \right.$ is continuous at $x = 0, $ then the value of $k$ is
✓

Answer

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

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