Discuss the continuity of the following functions at the indicate… - Vidyadip

Question

Discuss the continuity of the following functions at the indicated point:
$\text{f}\text{(x)}=\begin{cases}(\text{x}-\text{a}){\sin}\Big(\frac{1}{\text{x}-\text{a}}\Big) & \text{x} \neq \text{a}\\\ 0, & \text{ x} = \text{a}\end{cases}\text{at x}=\text{a}$

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Answer

We want, to check the continuity at x = 0.
$\text{LHL}=\lim\limits_{\text{x} \rightarrow 0^-}\text{f}\text{ (x)}=\lim\limits_{\text{h} \rightarrow 0}\text{f}(0-\text{ h)}=\lim\limits_{\text{x} \rightarrow 0}(-\text{h)}^2$
$\sin\Big(\frac{1}{\text{-h}}\Big)=0$
$\text{RHL}=\lim\limits_{\text{x} \rightarrow 0^+}\text{f}\text{ (x)}=\lim\limits_{\text{h} \rightarrow 0}\text{f}(0+\text{h)}=\lim\limits_{\text{x} \rightarrow 0}\text{h}^2$
$\sin\Big(\frac{1}{\text{h}}\Big)=0$
$\text{f}(0)=0$
Thus, LHL = RHL = f(0) = 0
Hence, the function is continuous at x = 0.

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