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Continuity — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsContinuity3 Marks
Question
$ \text {If } \mathrm{f}(\mathrm{x})=\frac{\sin 2 x}{5 x}-\mathrm{a}, \text { for } \mathrm{x}>0$
$ =4 \text { for } \mathrm{x}=0$
$=\mathrm{x}^2+\mathrm{b}-3, \text { for } \mathrm{x}<0 $
is continuous at $x=0$, find $a$ and $b$.
✓

Answer

$f(x)$ is continuous at $x=0$ (given)
$ \therefore  \lim _{x \rightarrow 0^{+}} \mathrm{f}(x)=\mathrm{f}(0)$
$\therefore  \lim _{x \rightarrow 0^{+}}\left(\frac{\sin 2 x}{5 x}-\mathrm{a}\right)=4$
$\therefore  \lim _{x \rightarrow 0^{+}} \frac{\sin 2 x}{5 x}-\lim _{x \rightarrow 0^{+}} \mathrm{a}=4$
$\therefore  \frac{1}{5} \lim _{x \rightarrow 0^{+}} \frac{\sin 2 x}{2 x} \times(2)-\lim _{x \rightarrow 0^{+}} \mathrm{a}=4$
$\therefore  \frac{1}{5}(1)(2)-\mathrm{a}=4 \quad[\because x \rightarrow 0,2 x \rightarrow 0$
$\therefore  \frac{2}{5}-\mathrm{a}=4 \quad\left[ \text { and } \lim _{\theta \rightarrow 0^{+}} \frac{\sin \theta}{\theta}=1 \right]$
$\therefore  \frac{2}{5}-4=\mathrm{a}$
$\therefore  \mathrm{a}=-\frac{18}{5}$
$\therefore \lim _{x \rightarrow 0^{-}}\left(x^2+\mathrm{b}-3\right)=4$
$\therefore b-3 =4$
$\therefore b=7$
$\therefore \frac{-18}{5}$ and $b=7$

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