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Question
Show that \[\lim_{x \to 2^-} \frac{x}{\left[ x \right]} \neq \lim_{x \to 2^+} \frac{x}{\left[ x \right]} .\]
Solution
\[\lim_{x \to 2^-} \frac{x}{\left[ x \right]}\]
\[\text{ Let } x = 2 - h, \text{ where h } \to 0 . \]
\[ \lim_{h \to 0} \frac{\left( 2 - h \right)}{\left[ 2 - h \right]}\]
\[ = \frac{2}{1}\]
\[ \lim_{x \to 2^+} \frac{x}{\left[ x \right]}\]
\[\text{ Let } x = 2 + \text{ h, where h } \to 0 . \]
\[ \lim_{h \to 0} \frac{\left( 2 + h \right)}{\left[ 2 + h \right]}\]
\[ = \frac{2}{2}\]
\[ = 1\]
\[ \therefore \lim_{x \to 2^-} \frac{x}{\left[ x \right]} \neq \lim_{x \to 2^+} \frac{x}{\left[ x \right]}\]
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