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Question
Show that \[\lim_{x \to 0} e^{- 1/x}\] does not exist.
Solution
\[\text{ Left hand limit }: \]
\[ \lim_{x \to 0^-} \left( e^{- \frac{1}{x}} \right)\]
\[\text{ Let } x = 0 - h, \text{ where h } \to 0 . \]
\[ \Rightarrow \lim_{h \to 0} e^\left( \frac{- 1}{0 - h} \right) \]
\[ = \lim_{h \to 0} \left( e^\frac{1}{h} \right)\]
\[ = \lim_{h \to 0} \left( e^\frac{1}{h} \right) \left[ \text{ When h } \to 0, then \frac{1}{h} \to \infty . \right]\]
\[ = e^\infty \]
\[ = \infty \]
\[\text{ Right hand limit }: \]
\[ \lim_{x \to 0^+} \left( e^{- \frac{1}{x}} \right)\]
\[\text{ Let } x = 0 + h, \text{ where h } \to 0 . \]
\[ \lim_{h \to 0} \left( e^\frac{- 1}{0 + h} \right)\]
\[ = \lim_{h \to 0} \left( \frac{1}{e^\frac{1}{h}} \right)\]
\[ = \frac{1}{\infty}\]
\[ = 0\]
\[ \lim_{x \to 0^-} e^{- \frac{1}{x}} \neq \lim_{x \to 0^+} e^\frac{1}{x} \]
\[\text{ Thus }, {limit}_{x \to 0} e^{- \frac{1}{x}} \text{ does not exist } .\]
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