Evaluate the following limits,if they exist,where [x] is the greatest interger function
a)lim [2x]/x as x approaches 0
b)lim x[1/x] as x approaches 0
a:x∈(−12,0):[2x]x=−1xlimx→0−[2x]x=−∞x∈(0,12):[2x]x=0limx→0+[2x]x=0The limit doesnt existb:x⋅(1x−1)<x[1x]⩽x⋅1xlimx→0x(1x−1)=1,limx→0x⋅1x=1⇒⇒limx→0x[1x]=1a:\\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\,\,doesnt\,\,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] =1a:x∈(−21,0):x[2x]=x−1x→0−limx[2x]=−∞x∈(0,21):x[2x]=0x→0+limx[2x]=0Thelimitdoesntexistb:x⋅(x1−1)<x[x1]⩽x⋅x1x→0limx(x1−1)=1,x→0limx⋅x1=1⇒⇒x→0limx[x1]=1
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