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

Answer

Given,
$\text{f}\text{(x)}=\text{x}\cos\Big(\frac{1}{\text{x}}\Big),\text{x}\neq0$
$\text{f}\text{(x)}=0,\ \text{x}=0$
We observe
$\lim\limits_{\text{x} \rightarrow 0}\text{f}\text{(x)}=\lim\limits_{\text{x} \rightarrow 0}\text{x}\cos\Big(\frac{1}{\text{x}}\Big)$
$\lim\limits_{\text{x} \rightarrow 0}\text{f}\text{(x)}=\lim\limits_{\text{x} \rightarrow 0}\text{x}\lim\limits_{\text{x} \rightarrow 0}\cos\Big(\frac{1}{\text{x}}\Big)$
$\lim\limits_{\text{x} \rightarrow 0}\text{f}\text{(x)}=0\times\lim\limits_{\text{x} \rightarrow 0}\cos\Big(\frac{1}{\text{x}}\Big)$
$=0$
$\lim\limits_{\text{x} \rightarrow 0}\text{f}\text{(x)}=\text{f}(0)$
Hence, f(x) is continuous at x = 0.

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