Answer
The limit does not exist in the following cases:
1.) When the limit is infinite as in the case of $f(x) = \frac{1}{|x-3|}$, which has infinite limit at $x=3$. See figure.
2.) When limit on the either side of the point is different. This is also called jump discontinuity as in the case of $f(x) =\frac{|x|}{x}, x\neq 0 $. It has a limit equal to 1 right side of 0 while limit equal to -1 on the left side of 0. See figure.
3.) The function oscillating around a point does not have limit at such a point. For instance, $f(x) = \sin\frac{1}{x}$ does not have limit at 0. See figure.
Work Step by Step
Any function does not have limit at a point when the denominator tends to 0 at that point. This is the first case. The functions having a break at some point are of second type and the oscillating functions are of third type.
