Question

How do you find the limit of `|x|/x` as x approaches `0`?

Answer 1

Does not exist

Explanation:

For `x < 0`, `(abs x)/x = (-x)/x = -1`

For `x >0`, `(abs x)/x = x/x = 1`

Thus
`lim_(x to 0^-) abs x/x = -1`

`lim_(x to 0^+) abs x/x = 1`

So the limit does not exist.

graph{|x|/x [-10, 10, -5, 5]}