Show that \[\lim_{x \to 0} \frac{x}{\left| x \right|}\] does not exist.
Solution
\[\lim_{x \to 0} \left( \frac{x}{\left| x \right|} \right)\]Left hand limit:
\[\lim_{x \to 0^-} \left( \frac{x}{\left| x \right|} \right) \]
\[\text{ Let } x = 0 - h, \text{ where } h \to 0 . \]
\[ \Rightarrow \lim_{h \to 0} \left( \frac{0 - h}{\left| 0 - h \right|} \right)\]
\[ = \lim_{h \to 0} \left( \frac{- h}{h} \right)\]
\[ = - 1\]
Right hand limit:
\[\lim_{x \to 0^+} \frac{\left( x \right)}{\left| x \right|}\]
\[\text{Let } x = 0 + h, \text{ where } h \to 0 . \]
\[ \lim_{h \to 0} \left( \frac{0 + h}{\left| 0 + h \right|} \right)\]
\[ = \lim_{h \to 0} \left( \frac{h}{h} \right)\]
\[ = 1\]
Left hand limit ≠ Right hand limit \[Thus, \lim_{x \to 0} \left( \frac{x}{\left| x \right|} \right) \text{ does not exist } .\]
