Is the following limit exist or not? $$\lim_{x\to 0} {\frac{\lfloor \frac{3}{2} +x\rfloor }{x}}$$
I have no idea about find right-hand and left-hand limits.
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2$\begingroup$ What is $[[ ]]$ $\endgroup$– Ahmed S. AttaallaJan 25, 2017 at 20:48
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1$\begingroup$ My first thought is the floor function, but that's clearly wrong... $\endgroup$– y_primeJan 25, 2017 at 20:50
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$\begingroup$ Yes, floor function. $\endgroup$– DomatesJan 25, 2017 at 20:51
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1 Answer
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The limit is not defined (it's infinity). Just fill in $x = 0$ and you get $\lim_{x\to0} \frac{\lfloor\frac{3}{2}+x\rfloor}{x} = \frac{\lfloor\frac{3}{2}+0\rfloor}{0} = \frac{1}{0}=\infty$. Note: for $x\to0$ with $x<0$, the limit is $-\infty$, for $x>0$ is it $+\infty$
You can also see this clearly when plotting the graph:
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$\begingroup$ Yes indeed, I was a bit too fast there :-) $\endgroup$– mitchbusJan 25, 2017 at 21:20