Why limits do not exist?

Question

lim x approaching 0 x / | x |

The back of the book states :

As x approaches 0 from the left, x / | x | approaches -1. As x approaches 0 from the right, x / | x | approaches 1. There is no single number L that the function values all get arbitrarily close to as x approaching 0.

I still don't understand this answer. Can someone clarify it for me? What is another way of answering this question? Can't we just state anything divided by zero is undefined?

Answer 1

The problem here is not that the limit is undefined, but rather that teh function is multivalued at x = 0. There is a discontinuity at x = 0. There is no asymptote.

x/|x| consists of two disconnected lines. That's why the limit at x = 0 does not exist. Just to test it out, work out x/|x| for x = -0.001 and x = +0.001. See what you get.

Just as a side note, if you obtained the Fourier series for x/|x|, it would return the value 0 at x = 0. That's the average of 1 and -1.

Answer 2

Well, you gave the answer. Exactly what you said. In order for a function to have a limit at x =a, in your case a =0, then the limit when x approaches a from the right must equal the limit when x approaches a from the left. In this case, the limit from the left is -1 and the limit from the right is 1. So, the only candidates to limit of f at 0 are -1 and 1. But when you approach 0 from the left, the function gets "far" from 1, so that 1 cannot be a limit at x =0. And when you approach 0 from the right, the function gets "far" from -1, so that -1 canot be a limit either.

It follows f has no limit at 0. And no, you can't conclude this stating anything divided by zero is undefined. This is true, but when you take the limit of a function at a point a, you're interested in the behavior of the function in a neighborhood of a but excluding a. The value of f at a, or even the existence of f(a) are irrelevant for the existence or for the value of the limit at a.

Answer 3

Not everything divided by zero doesn't exist

for instance
(x+2)(x-2)/(x-2) approaches a limit of 4 as x => 2

But in your case, depending on the direction in which you approach, the answer is different. Therefore the limit does not exist. This means that your function is not continuous at 0.

Answer 4

For the general limit to exist, the right handed limit and the left handed limit must be equal. Since they are 1 and -1, they are not equal and so there is no general limit.

Anything divided by zero is not necessarily undefined. 0/0 is considered "indeterminate" rather than "undefined" because its value depends on the function, as demonstrated by IamSpazzy.

Answer 5

In this case, a limit does exist, actually, two of them. However, you must specify how you approach that limit to get an unambiguous number. If you don't specify, you must deal with this plural answer and have no reasonable way to choose between the two possible values.