Question

\[\large \lim\limits_{x\to 0^{+}} \left[ x\left[\dfrac{1}{x}\right]\right] = ?\]

Answer 1

[] is the greatest integer function

Answer 2

thanks ;)
i supposed \( \large y = \frac{1}{x} \) and when \(x \rightarrow 0^+\) then \(y \rightarrow \infty\).
\(\Large \frac{ 1 }{ y} \left[ y \right]-1 \le \left[ \frac{ 1 }{ y }\left[ y \right] \right] \le \frac{ 1 }{ y }\left[ y \right]\)

Answer 3

Thats a clever way to apply squeeze thm xD

Answer 4

if we suppose \(\left\{ y \right\} = y - \left[ y \right] \) and \(0 \le \left\{ y \right\} < 1\)

then we can say,
\(\large - \frac{ \left\{ y \right\} }{ y } \le \left[ \frac{ 1 }{ y }\left[ y \right] \right] \le -\frac{ \left\{ y \right\} }{ y } + 1\)

Answer 5

@ganeshie8 no not very clever ;D , if we get Limit when y approaches \( \infty \) then we would get a bad thing :|

Answer 6

we have this
\[x - 1 \lt \left[x\right]\le x\]

Answer 7

\(\large -\frac{ 1 }{ y } < - \frac{ \left\{ y \right\} }{ y } \le 0\)

Answer 8

yes we have that! ans this means we can write it as \( \le \) because \( x < a \) then \( x-1 \le a \)

Answer 9

\[\large \lim\limits_{x\to 0^{+}} \left[ x\left[\dfrac{1}{x}\right]\right] = \lim\limits_{y\to \infty } \left[ \dfrac{[y]}{y}\right] \]

Answer 10

So do we get \[\large \lim\limits_{y\to \infty } \dfrac{[y]}{y} - 1\lt \lim\limits_{y\to \infty } \left[ \dfrac{[y]}{y}\right] \le \lim\limits_{y\to \infty } \dfrac{[y]}{y}\]

?

Answer 11

yes i think we've solved \(\large \lim_{y \rightarrow \infty } \frac{ \left[ y \right] }{ y }\)

Answer 12

yes that equals 1

Answer 13

\[\large \lim\limits_{y\to \infty } \dfrac{[y]}{y} - 1\lt \lim\limits_{y\to \infty } \left[ \dfrac{[y]}{y}\right] \le \lim\limits_{y\to \infty } \dfrac{[y]}{y}\]

evaluating the limits gives

\[\large 1 - 1\lt \lim\limits_{y\to \infty } \left[ \dfrac{[y]}{y}\right] \le 1\]

\[\large 0 \lt \lim\limits_{y\to \infty } \left[ \dfrac{[y]}{y}\right] \le 1\]

Answer 14

yeah not so much useful :O

Answer 15

but,
\(\Large 0 < \lim_{y \rightarrow \infty } \left[ \frac{ \left[ y \right] }{ y } \right] \rightarrow 1 \le \lim_{y \rightarrow \infty } \left[ \frac{ \left[ y \right] }{ y } \right]\)

Answer 16

@ganeshie8 what do u think?!

Answer 17

yeah we can conclude that the limit is between 0 and 1 but how good is that ?

plugin x = 0.1
[0.1*[1/0.1]] = [0.1*10] = [1] = 1

but look at the graph :
http://i.gyazo.com/0b2f34a6373523f390ea15308b946140.png

Answer 18

so the answer would be 0 if x approaches \( x^+ \)

Answer 19

* \( 0 ^+ \)

Answer 20

consider a small positive number \(\large x = 10^{-100000}\)

Answer 21

\[\left[ x\left[\dfrac{1}{x}\right]\right] = \left[ 10^{-10000}\left[\dfrac{1}{10^{-100000}}\right]\right] \]

Answer 22

\[ = \left[ 10^{-10000}\left[10^{100000}\right]\right] \]

Answer 23

\[ = \left[ 10^{-10000}\cdot 10^{100000}\right] \]

Answer 24

\[ = \left[ 10^{0}\right] \]

Answer 25

\[ = 1\]

Answer 26

I don't understand why the graph is showing 0 for x >0 :/

Answer 27

https://www.desmos.com/calculator/iij7exbkxs

Answer 28

well,forget it ;) it's not related to this limit,will ask later.

Answer 29

@ganeshie8 , i think i have found the mistake...why \(\large \lim_{y \rightarrow \infty } \frac{ \left[ y \right] }{ y } = 1\) i know u have shown it before but i think u had shown it in correctly!

Answer 30

\(\large 0 \le \frac{ \left\{ y \right\} }{ y } < \frac{ 1 }{ y }\) so we ca't use squeeze thm!

Answer 31

but to my surprise the answer of your example!why?

Answer 32

[1/x] integer thus the limit is the same as
lim ([x][1/x])
sonsider peicewise function



At the end i say the LIMIT DOES NOT EXISTS