In what situations does a limit not exist?

Limits typically fail to exist for one of four reasons: The one-sided limits are not equal. The function doesn't approach a finite value (see Basic Definition of Limit). The function doesn't approach a particular value (oscillation).

Why would a limit not exist?

In short, the limit does not exist if there is a lack of continuity in the neighbourhood about the value of interest. ... Most limits DNE when limx→a−f(x)≠limx→a+f(x) , that is, the left-side limit does not match the right-side limit.