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Continuity — Maths STD 11 Science — Question

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

Answer

Correct option: A.
0
(a) : $f(x)=\left\{\begin{array}{cc}\log \left(\sec ^2 x\right)^{\cot ^2 x}, & x \neq 0 \\ K+1 & , x=0\end{array}\right.$ is continuous at $x=0$.
$
\begin{aligned}
& \Rightarrow \lim _{x \rightarrow 0} f(x)=f(0) \Rightarrow \lim _{x \rightarrow 0} \log \left(\sec ^2 x\right)^{\cot ^2 x}=K+1 \\
& \Rightarrow \lim _{x \rightarrow 0} \cot ^2 x \log \left(1+\tan ^2 x\right)=K+1 \\
& \Rightarrow \lim _{x \rightarrow 0} \frac{\log \left(1+\tan ^2 x\right)}{\tan ^2 x}=K+1 \\
& \Rightarrow K+1=1 \Rightarrow K=0 \quad \quad\left[\because \lim _{x \rightarrow 0} \frac{\log (1+x)}{x}=1\right]
\end{aligned}
$

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