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Continuity — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsContinuity2 Marks
Question
If $\text{f(x)}=\begin{cases}\frac{\sin^{-1}\text{x}}{\text{x}},&\text{x}\neq0\\\text{k},&\text{x}=0\end{cases}$ is continuous at x = 0, write the value of k.
✓

Answer

Given, $\text{f(x)}=\begin{cases}\frac{\sin^{-1}\text{x}}{\text{x}},&\text{x}\neq0\\\text{k},&\text{x}=0\end{cases}$
If f(x) is continuous at x = 0, then
$\lim\limits_{{\text{x}}\rightarrow0}\text{f(x})=\text{f}(0)$
$\Rightarrow\lim\limits_{{\text{x}}\rightarrow0}\text{f(x})\Big(\frac{\sin^{-1}\text{x}}{\text{x}}\Big)=\text{f}(0)$
$\Rightarrow\lim\limits_{{\text{x}}\rightarrow0}\Big(\frac{\sin^{-1}\text{x}}{\text{x}}\Big)=\text{k}$
$\Rightarrow\text{k}=1$ $\bigg[\because\ \lim\limits_{{\text{x}}\rightarrow0}\Big(\frac{\sin^{-1}\text{x}}{\text{x}}\Big)=1\bigg]$

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