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Question
\[\lim_{x \to 1} \left[ x - 1 \right]\] where [.] is the greatest integer function, is equal to
Options
1
2
0
does not exist
Solution
We have,
\[\lim_{x \to 1^-} \left[ x - 1 \right] = \lim_{h \to 0} \left[ 1 - h - 1 \right] = \lim_{h \to 0} \left[ - h \right] = - 1\]
Also,
\[\lim_{x \to 1^+} \left[ x - 1 \right] = \lim_{h \to 0} \left[ 1 + h - 1 \right] = \lim_{h \to 0} \left[ h \right] = 0\]
\[\therefore \lim_{x \to 1^-} \left[ x - 1 \right] \neq \lim_{x \to 1^+} \left[ x - 1 \right]\]
Thus,
\[\lim_{x \to 1} \left[ x - 1 \right]\] does not exist.
Hence, the correct answer is option (d).
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