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Why the limit of this greatest integer function doesn't exist: $$\lim_{x\to0}[x]$$

My attempt:
L.H.L.= $$\lim _{x\to0^-}[x]$$ $$=\lim_{h\to0}[0-h]$$ $$=\lim_{h\to0}[0-0]$$ $$=\lim_{h\to0}[0]$$ $$=0$$ R.H.L.= $$\lim _{x\to0^+}[x]$$ $$=\lim_{h\to0}[0+h]$$ $$=\lim_{h\to0}[0+0]$$ $$=\lim_{h\to0}[0]$$ $$=0$$

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    Your problem is that you're applying the limit within the $[x]$ function when the question is specifically asking you to prove that you can't do that. – 2017-01-18

1 Answers 1

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You have to solve the greatest integer function with the help of limit first.
L.H.L.= $$\lim _{x\to0^-}[x]$$ $$=\lim_{h\to0}[0-h]$$ $$=\lim_{h\to0}-1\tag{since h is a postive quantity}$$ $$=-1$$ R.H.L.= $$\lim _{x\to0^+}[x]$$ $$=\lim_{h\to0}[0+h]$$ $$=\lim_{h\to0}0\tag{since h is a positive quantit}$$ $$=0$$

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    Nice answer. +1 – 2017-01-18