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.
$\lim_{x\to 0} {\frac{\lfloor \frac{3}{2} +x\rfloor }{x}}$
0
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calculus
limits
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2What is $[[ ]]$ – 2017-01-25
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1My first thought is the floor function, but that's clearly wrong... – 2017-01-25
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0Yes, floor function. – 2017-01-25
1 Answers
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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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2Isn't $\lfloor 3/2+0\rfloor =1$? – 2017-01-25
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0Yes indeed, I was a bit too fast there :-) – 2017-01-25