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CONTINUITY AND DIFFERENTIABILITY — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsCONTINUITY AND DIFFERENTIABILITY5 Marks
Question
Find which of the function:
$\text{f(x)}=\begin{cases}|\text{x}|\cos\frac{1}{\text{x}},&\text{if x}\neq0\\0,&\text{if x}=0\end{cases}$
at x = 0
✓

Answer

We have, $\text{f(x)}=\begin{cases}|\text{x}|\cos\frac{1}{\text{x}},&\text{if x}\neq0\\0,&\text{if x}=0\end{cases}$ at x = 0
At x = 0, $\text{L.H.L}=\lim\limits_{\text{h}\rightarrow0}|\text{x}|\cos\frac{1}{\text{x}}$
$=\lim\limits_{\text{h}\rightarrow0}|0-\text{h}|\cos\frac{1}{0-\text{h}}$
$=\lim\limits_{\text{h}\rightarrow0}\text{h}\cos\frac{1}{\text{h}}$
= 0 × [an oscillating number between -1 and 1] = 0
$\text{R.H.L}=\lim\limits_{\text{x}\rightarrow0^+}|\text{x}|\cos\frac{1}{\text{x}}$
$=\lim\limits_{\text{h}\rightarrow0}|0+\text{h}|\cos\frac{1}{0+\text{h}}$
$=\lim\limits_{\text{h}\rightarrow0}\text{h}\cos\frac{1}{\text{h}}$
= 0 × [an oscillating number between -1 and 1] = 0
Also f(0) = 0 (given)
Thus, L.H.L = R.H.L = f(0)
Hence, f(x) is discontinuous at x = 0.

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