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Question
Discuss the continuity of the following functions at the indicated point(s):
(i) \[f\left( x \right) = \begin{cases}\left| x \right| \cos\left( \frac{1}{x} \right), & x \neq 0 \\ 0 , & x = 0\end{cases}at x = 0\]
Solution
(i) Given:
We observe
\[\lim_{x \to 0} f\left( x \right) = \lim_{x \to 0} \left| x \right| \cos\left( \frac{1}{x} \right)\]
\[ \Rightarrow \lim_{x \to 0} f\left( x \right) = \lim_{x \to 0} \left| x \right| \lim_{x \to 0} \cos\left( \frac{1}{x} \right)\]
\[ \Rightarrow \lim_{x \to 0} f\left( x \right) = 0 \times \lim_{x \to 0} \cos\left( \frac{1}{x} \right) = 0\]
Hence, f(x) is continuous at x = 0.
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