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Continuity — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsContinuity2 Marks
MCQ
If $f(x)=\left\{\begin{array}{cc}\log \left(\sec ^2 x\right)^{\cot ^2 x} & \text { for } x \neq 0 \\ K & \text { for } x=0\end{array}\right.$ is continuous at $x=0$ then $K$ is
  • A
    $e^{-1}$
  • ✓
    1
  • C
    $e$
  • D
✓

Answer

Correct option: B.
1
(b) : Since $f(x)$ is continuous at $x=0$.
$\therefore f(0)=\lim _{x \rightarrow 0} \log \left(\sec ^2 x\right)^{\cot ^2 x}$
$\Rightarrow K=\lim _{x \rightarrow 0} \cot ^2 x \cdot \log \left(1+\tan ^2 x\right) \Rightarrow K=\lim _{x \rightarrow 0} \frac{\log \left(1+\tan ^2 x\right)}{\tan ^2 x}$
$\Rightarrow K=1 \quad\left[\because \lim _{x \rightarrow 0} \frac{\log (1+x)}{x}=1\right]$

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