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Question
Evaluate \[\lim_{x \to 0} f\left( x \right)\] where \[f\left( x \right) = \begin{cases}\frac{\left| x \right|}{x}, & x \neq 0 \\ 0, & x = 0\end{cases}\]
Solution
\[f\left( x \right) = \begin{cases}\frac{\left| x \right|}{x}, & x \neq 0 \\ 0, & x = 0\end{cases}\]
\[LHL: \]
\[ \lim_{x \to 0^-} \left( \frac{\left| x \right|}{x} \right)\]
\[\text{ Let } x = 0 - h, \text{ where } h \to 0 . \]
\[ \Rightarrow \lim_{h \to 0} \left( \frac{\left| 0 - h \right|}{- h} \right)\]
\[ = \lim_{h \to 0} \left( \frac{h}{- h} \right) = - 1\]
\[RHL: \]
\[ \lim_{x \to 0^+} f\left( x \right)\]
\[ = \lim_{x \to 0^+} \left( \frac{\left| x \right|}{x} \right)\]
\[\text{ Let } x = 0 + h, \text{ where } h \to 0 . \]
\[ \lim_{h \to 0} \left( \frac{\left| 0 + h \right|}{0 + h} \right) = 1\]
\[LHL \neq RHL\]
\[ \text{ Thus, \lim}_{x \to 0} f\left( x \right) \text{ does not exist } .\]
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