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5. continuity and differentiation — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths5. continuity and differentiation1 Mark
MCQ
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
  • ✓
    $1$
  • B
    $-1$
  • C
    $0$
  • D
    $e$
✓

Answer

Correct option: A.
$1$
a
(a) $\mathop {\lim }\limits_{x \to 0} \,\,{(\cos x)^{1/x}} = k\,\, \Rightarrow \,\,\mathop {\lim }\limits_{x \to 0} \frac{1}{x}\log \,(\cos x) = \log k$

$ \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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