Question

Why doesn't the limit for |x| exist?

Answer 1

x--->0 I assume?

Answer 2

As x--->0 is the only place where that function has a limit that does not exist.

Answer 3

@Easyaspi314 the limit of |x| as x approaches 0 is 0...

Answer 4

I think so as well

Answer 5

Left and right hand limits are both 0. It's not differentiable at x=0, but that doesn't mean the limit doesn't exist. Wolfram alpha agrees that it exists. http://www.wolframalpha.com/input/?i=limit+of+ABS%28x%29+as+x+approaches+0

Answer 6

|x| is a corner, so at 0, the limit DNE. My question is why doesn't it exist?

Answer 7

Corners don't make limits fail to exist. They make them not differentiable.

Answer 8

Another example:
http://www.uiowa.edu/~examserv/mathmatters/tutorial_quiz/geometry/piecewisefunctions_clip_image010.gif

As x approaches -1, the limit of this function is 1. Why would the corner affect that?

Answer 9

Conditions for a limit to exist:

The limit from the left must exist.
The limit from the right must exist.
The limits from both directions must be equal.

Show me how\[\Large \lim_{x \rightarrow 0} \left| x \right|\] fails. And again, here's wolfram alpha showing it exists: http://www.wolframalpha.com/input/?i=limit+of+ABS%28x%29+as+x+approaches+0