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Continuity — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsContinuity2 Marks
Question
If $\text{f(x)}=\begin{cases}\frac{\text{x}}{\sin3\text{x}},&\text{x}\neq0\\\text{k},&\text{x}=0\end{cases}$ is continuous at x = 0, then write the value of k.
✓

Answer

If f(x) is continuous at x = 0, then
$\lim\limits_{{\text{x}}\rightarrow0}\text{f(x})=\text{f}(0)$
$\Rightarrow\lim\limits_{{\text{x}}\rightarrow0}\frac{\text{x}}{\sin3\text{x}}=\text{k}$
$\Rightarrow\lim\limits_{{\text{x}}\rightarrow0}\frac{1}{\frac{\sin3\text{x}}{\text{x}}}=\text{k}$
$\Rightarrow\lim\limits_{{\text{x}}\rightarrow0}\frac{1}{\frac{3\sin3\text{x}}{3\text{x}}}=\text{k}$
$\Rightarrow\frac{1}{3}\Bigg(\frac{1}{\lim\limits_{{\text{x}}\rightarrow0}\frac{3\sin3\text{x}}{3\text{x}}}\Bigg)=\text{k}$
$\Rightarrow\text{k}=\frac{1}{3}$

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