Question

How do you know a limit does not exist?

Answer 1

In short, the limit does not exist if there is a lack of continuity in the neighbourhood about the value of interest.

Recall that there doesn't need to be continuity at the value of interest, just the neighbourhood is required.

Most limits DNE when `lim_(x->a^-)f(x)!=lim_(x->a^+)f(x)`, that is, the left-side limit does not match the right-side limit. This typically occurs in piecewise or step functions (such as round, floor, and ceiling).

A common misunderstanding is that limits DNE when there is a point discontinuity in rational functions. On the contrary, the limit exists perfectly at the point of discontinuity!

So, an example of a function that doesn't have any limits anywhere is `f(x) = {x=1, x in QQ; x=0, otherwise}`. This function is not continuous because we can always find an irrational number between 2 rational numbers and vice versa.