Question

what is the limit of (x-1)/(|x-1|) as x approaches 1?

Answer 1

The limit here will not exist.

Answer 2

Do you know why?

Answer 3

is it because there is an abs value on the bottom?

Answer 4

does that mean that there will never be a limit when there is an abs value?

Answer 5

Well, as x begins to converge on the constant, yes. Because in this case, when you take the left and right handed limits, the signs will change in the numerator, however they will NOT change in the denominator.

You get a jump discontinuity.

Answer 6

IE: the left and right sided limits do not agree

Answer 7

If, however, the limit were "as x approaches 2", well then the limit would exist.

Answer 8

ohhh because it would equal zero either way? the denominator

Answer 9

Well, if you look at the numerator, you have a case where you have a value just a smidge smaller than 1, this makes the numerator negative. In the right sided limit, you get a value just a smidge larger than 1. So it's positive. However, in the denominator this does not happen. It's an absolute value. So the right and left sided limits do not agree. The left approaches -1, the right approaches 1. This is a jump discontinuity.

Answer 10

So the limit DNE

Answer 11

ohhhhhh okay thank you!