2

I have this limit, which ends up as $\lim_{x\to0}\frac{|x|}{x}$ which yields $\frac00$.

Normally, one would say that this limit doesn't exist, but at the same time, we have L'Hopital's rule which often times can deal with this.

My question is: How do I know for certain that the limit doesn't exist? And I'm talking in general here, not just for the above-mentioned example. I mean, some times, you can do L'Hopital 3 times, still giving 0/0, but on the 4th time, you get a defined value. Where does one draw the line and say "no, it doesn't exist"?

Alec
  • 4,094
  • I'm not sure how general it's possible to get, but in this case you can look at the limit as $x$ tends to zero through positive and negative values separately. You will find that you get two different answers. That is, the function approaches different values if $x$ approaches $0$ from the right or the left, so if we just want to say "$x$ approaches zero" without specifying right or left, then the limit does not exist. – David Jun 21 '14 at 05:46
  • Before applying any machinery, it is useful to look at the function. It would be nice if the denominator did not approach $0$. But unfortunately it does. So let us look at the behaviour of the function near $0$. The answer is clear, the function is identically $1$ for $x$ positive, and identically $-1$ for $x$ negative. – André Nicolas Jun 21 '14 at 05:47
  • Sorry but how does L'Hopital's rule enter the picture here? – Did Jun 21 '14 at 05:47
  • About the (separate) question of when to go on with L'Hospital's Rule: when there are clear signs of progress. – André Nicolas Jun 21 '14 at 05:50
  • 1
    You are incorrect to say "normally, a limit of the form $0/0$ doesn't exist". The form $0/0$ is an indeterminate one, meaning that it doesn't tell you what the limit is. In fact, it's as bad as an indeterminate form can possibly be, since it gives you no information whatsoever about what the limit can be or if it exists. (The algebraic expression $0/0$ is undefined, but that only has an indirect relation to any of the above: specifically, the fact the form $0/0$ is indeterminate is a compelling reason why we should not define a value for the algebraic expression $0/0$) –  Jun 21 '14 at 06:33
  • @Hurkyl - Good point! I guess what I meant to say was that when one encounters the 0/0 phenomenon, one tends to conclude that the limit leading to it doesn't exist. I guess that would be a wrong assumption. Then it's sad to say that so many learning resources, particularly online, tend to draw this conclusion. – Alec Jun 21 '14 at 08:16

1 Answers1

4

As for real $x, |x|=+\sqrt{x^2}$

$$\lim_{x\to 0^-}\frac{ |x|}x=\lim_{x\to 0^-}\frac{-x}x=-1$$ as $x\to0\implies x\ne0$

Similarly, $$\lim_{x\to 0^+}\frac{ |x|}x=+1$$

So, the left & the right are not same

lab bhattacharjee
  • 274,582