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What is the limit of

$$\lim_{x\rightarrow 0} \frac{x}{|x|}$$

My guess is that it is $1$ if approached from $0+$ and $-1$ if approached from $0-$

Thanks

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    $\begingroup$ So basically the limit doesn't exist. $\endgroup$
    – S.C.B.
    Mar 12, 2016 at 16:52
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    $\begingroup$ Good guess! So, how can you write rigorously what you 'know' intuitevely? Using the definition of the limit what does this mean here? Make a try. $\endgroup$ Mar 12, 2016 at 16:52
  • $\begingroup$ Well I am not sure how to show it mathematically rigorously, because L'Hospital's rule just turns the fraction upside down so you can't really guess from that. But graphically it makes sense. Would appreciate a hint here. But I have a more urgent problem related to this. If the limits can be evaluated in this way what is then: $\int \frac{x}{|x|}\delta(x)dx$ $\endgroup$
    – onephys
    Mar 12, 2016 at 16:54
  • $\begingroup$ @onephys You don't need L'Hospital, I think. Divide the cases. $\endgroup$
    – S.C.B.
    Mar 12, 2016 at 16:56
  • $\begingroup$ L'Hospital's Rule does not make it inconclusive. You cannot use the Rule unless you first apply the definition of the absolute value. You then must observe the right hand limit and the left hand limit separately to find 1 and -1 which is confirmed by the graph. Hence as indicated earlier, THE limit does not exist. $\endgroup$
    – imranfat
    Mar 12, 2016 at 16:57

2 Answers 2

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we should discuss two cases for this question (according to definition of modulus function)

if $x>0$ the function is $$\lim_{x\to 0+}\frac{x}{x}, x>0$$ LHL

$$\implies\lim_{x\to 0+}1=1$$

if $x<0$ the function is $$\lim_{x\to 0-}\frac{x}{-x}, x<0$$

RHL

$$\lim_{x\to 0-}-1=-1$$

as you can see left hand limit is not equal to right hand limit. So limit doesn't exist!!

Note:

the + and - signs in limits

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As $\ x \rightarrow 0 $ it can approach it from negative side and positive side.
For
Left Hand Limit: $\ x \rightarrow 0- $
i.e, $x<0$
Hence $|x|$ = $-x$
So $\lim_{x\rightarrow 0-} \ x/|x|$ = $\lim_{x\rightarrow 0-} \ x/(-x)$ = $-1$
Similarly for
Right Hand Limit: $\ x \rightarrow 0+$
Now $\lim_{x\rightarrow 0+} \ x/|x|$ = $\lim_{x\rightarrow 0-} \ x/(x)$ = $+1$
Since
RHL $\neq$ LHL
Therefore Limit doesn't exist.

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