Answer to Question #315584 in Calculus for Aune

Question

Evaluate the following limits, if they exist, where ⌊𝑥⌋ is the greatest integer function.

(a)lim ⌊2𝑥⌋/𝑥

𝑥→0

(b) lim 𝑥 ⌊1/𝑥⌋

𝑥→0

Answer 1

$a:\\x\in \left( -\frac{1}{2},0 \right) :\frac{\left[ 2x \right]}{x}=\frac{-1}{x}\\\underset{x\rightarrow 0-}{\lim}\frac{\left[ 2x \right]}{x}=-\infty \\x\in \left( 0,\frac{1}{2} \right) :\frac{\left[ 2x \right]}{x}=0\\\underset{x\rightarrow 0-}{\lim}\frac{\left[ 2x \right]}{x}=0\\The\,\,\lim it\,\,doesn‘t\,\,exist\\b:\\x\cdot \left( \frac{1}{x}-1 \right) <x\left[ \frac{1}{x} \right] \leqslant x\cdot \frac{1}{x}\\\underset{x\rightarrow 0}{\lim}x\left( \frac{1}{x}-1 \right) =1,\underset{x\rightarrow 0}{\lim}x\cdot \frac{1}{x}=1\Rightarrow \\\Rightarrow \underset{x\rightarrow 0}{\lim}x\left[ \frac{1}{x} \right] =1$