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Continuity — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsContinuity2 Marks
MCQ
$\begin{array}{lc}\text {Let } f (x)=\left\{\begin{array}{rc}(1+|\sin x|)^{\frac{1}{|\sin x|}}, & -\frac{\pi}{6} < x<0 \\ b, & x=0 \\ e ^{\frac{\tan 2 x}{\tan 3 x}}, & 0 < x<\frac{\pi}{6}\end{array}\right.  \end{array}$
Then the values of $a$ and $b$ if $f$ is continuous at $x=0$, are respectively
  • A
    $\frac{2}{3}, \frac{3}{2}$
  • ✓
    $\frac{2}{3}, e ^{\frac{2}{3}}$
  • C
    $\frac{3}{2}, e ^{\frac{3}{2}}$
  • D
    none of these
✓

Answer

Correct option: B.
$\frac{2}{3}, e ^{\frac{2}{3}}$
(B)
For $f (x)$ to be continuous at $x=0$, we must have
$\lim _{x \rightarrow 0^{-}} f (x)= f (0)=\lim _{x \rightarrow 0^{+}} f (x)$
$\lim _{x \rightarrow 0^{+}} f (x)=\lim _{x \rightarrow 0^{+}} e ^{\tan 2 x / \tan 3 x}$
$=\lim _{x \rightarrow 0^{+}} e ^{\left(\frac{\tan 2 x}{2 x} \times 2 x\right) /\left(\frac{\tan 3 x}{3 x} \times 3 x\right)}$
$= e ^{\frac{2}{3}}$
$f (0)=\lim _{x \rightarrow 0^{+}} f (x)$
$\Rightarrow b = e ^{\frac{2}{3}}$
$\lim _{x \rightarrow 0^{-}} f (x)=\lim _{x \rightarrow 0^{-}}(1+|\sin x|)^{ a /|\sin x|}$
$= e ^{\lim _{x \rightarrow 0}\left(|\sin x| \times \frac{ a }{|\sin x|}\right)}= e ^{ a }$
$f (0)=\lim _{x \rightarrow 0^{-}} f (x)$
$\Rightarrow b = e ^{ a } \Rightarrow e ^{\frac{2}{3}}= e ^{ a }$
$\Rightarrow a =\frac{2}{3}$

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