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Question
Find: \[ \lim_{x \to 1} \left[ x \right]\]
Solution
\[\text{ LHL }: \]
\[ \lim_{x \to 1^-} \left[ x \right]\]
\[\text{ Let } x = 1 - h, \text{ where h } \to 0 . \]
\[ \lim_{h \to 0} \left[ 1 - h \right]\]
\[ = 0\]
\[\text{ RHL }: \]
\[ \lim_{x \to 1^+} \left[ x \right]\]
\[\text{ Let } x = 1 + h, \text{ where h } \to 0 . \]
\[ \lim_{h \to 0} \left[ 1 + h \right] = 1\]
\[ LHL \neq RHL\]
\[\text{ Thus }, \lim_{x \to 1} \left[ x \right] \text{ does not exist } .\]
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