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Continuity — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsContinuity3 Marks
Question
Test the continuity of the function on f(x) at the origin:
$\text{f}\ (\text{x})=\begin{cases}\frac{\text{x}}{\text{|x|}},& \text{x}\neq0\\1, & \text{x} = 0\end{cases}$
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Answer

Given,
$\text{f}\ (\text{x})=\text{x},\text{ x}\neq0$
$\text{f}\ (\text{x})=1,\text{ x}=0$
We observe
$\text{(LHL at x}= 0)$
$\lim\limits_{\text{x} \rightarrow 0^-}\text{f}\text{ (x)}=\lim\limits_{\text{h} \rightarrow 0}\text{f} \ (0-\text{h})$
$\lim\limits_{\text{x} \rightarrow 0^-}\text{f}\text{ (-h)}=\lim\limits_{\text{h} \rightarrow 0} \frac{\text{-h}}{\text{h}}$
$\lim\limits_{\text{h} \rightarrow 0}-1=-1$
$\text{(RHL at x}=0)$
$\lim\limits_{\text{x} \rightarrow 0^+}\text{f}\ \text{(x)}=\lim\limits_{\text{h} \rightarrow 0}\text{f}(0+\text{h})$
$\lim\limits_{\text{h} \rightarrow 0}\text{f}\ \text{(h)}=\lim\limits_{\text{h} \rightarrow 0}\frac{\text{h}}{\text{h}}$
$\lim\limits_{\text{h} \rightarrow 0}1=1$
Hence, f(x) is discontinuous at the origin.

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