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Question
Prove that \[\lim_{x \to a^+} \left[ x \right] = \left[ a \right]\] R. Also, prove that \[\lim_{x \to 1^-} \left[ x \right] = 0 .\]
Solution
\[\lim_{x \to a^+} \left[ x \right]\]
\[\text{ Let } x = a + h, \text{ where h } \to 0 . \]
\[ \lim_{h \to 0} \left[ a + h \right]\]
\[ = a\]
\[ \lim_{x \to 1^-} \left[ x \right]\]
\[\text{ Let } x = 1 - \text{ h, where h } \to 0 . \]
\[ \lim_{h \to 0} \left[ 1 - h \right]\]
\[ = 0\]
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