Let $f(x) = \lfloor x\lfloor1/x\rfloor \rfloor $ . Find $\lim_{x \to 0^{+} } f(x) $ and $\lim_{x \to 0^{} } f(x)$ . I think $\lim_{x \to 0^{+} } f(x)$ doesn't exist but I have no idea about $\lim_{x \to 0^{} } f(x)$ .
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If $1/x$ is an integer then $x\lfloor 1/x\rfloor=1$. If $1/x$ is not an integer, and $x>0$, then $x\lfloor 1/x\rfloor<1$. So the righthand limit doesn't exist.
If $x<0$ and $1/x$ isn't an integer then $x\lfloor 1/x\rfloor>1$. However, $x\lfloor 1/x\rfloor<x(1/x1)=1x$, so provided $x>1$ we have $x\lfloor 1/x\rfloor<2$.
Especially Lime
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Can you explain about the lefthand limit ? – S.H.W Dec 21 '17 at 15:34

So the lefthand limit is $1$ ? – S.H.W Dec 21 '17 at 15:59

Yes, in fact $f(x)\equiv 1$ on the interval $(1,0)$. (It's also $1$ on the interval $(2,1)$ but for a different reason.) – Especially Lime Dec 21 '17 at 17:12
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Check the following graph:
You will get the answer automatically.
https://www.desmos.com/calculator/8p190y7wr2
Hint: The limit as a whole is not defined, because negative limit is not equal to positive limit.
Anivarth
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2Thanks but sometimes graphing programs can lead us to the wrong answer . For example here if you consider $a_n = 1/n$ and $1/{(n+0.5)}$ then you will notice that $\lim_{n \to \infty} f(a_n) = 1$ and $\lim_{n \to \infty} f(b_n) = 0$ . Therefore the righthand limit doesn't exist but you can't see that from the graph . – S.H.W Dec 21 '17 at 12:41

The Desmos graph plots all values correctly (at least in the vicinity of $x=0$) _except_ when $x$ is the reciprocal of a positive integer. But it's those values that cause the limit as $x\to0+$ to be undefined. – David K Dec 21 '17 at 13:37

Graphs are useful only for simple functions and for the current example it does not plot the points for $x=1/n$ where $n$ is a positive integer. – Paramanand Singh Dec 21 '17 at 19:42