Question
Evaluate the following limits:
$\lim _{x \rightarrow 0}[x]$ ([] is a greatest integer function.)

Answer

$ \lim _{x \rightarrow 0}[x]$
${[x]=-1 \quad ;-1 \leq x<0}$
${[x]=0 \quad ; 0 \leq x<1}$
$\lim _{x \rightarrow 0^{-}}[x]=-1$
$\lim _{x \rightarrow 0^{+}}[x]=0$
$\therefore \quad \lim _{x \rightarrow 0^{-}}[x] \neq \lim _{x \rightarrow 0^{+}}[x]$
$\therefore \quad \lim _{x \rightarrow 0}[x] \text { does not exist. }$

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