Indeterminate forms

Some singularities are easy to diagnose. Consider the function $\frac{\cos x}{x}$ at the point $x = 0$. The function evaluates to $\frac{1}{0}$ and is thus discontinuous at that point. Since the numerator and denominator are continuous functions and the denominator vanishes while the numerator does not, the left and right limits as $x \to 0$ do not exist. Thus the function has an infinite discontinuity at the point $x = 0$.

Figure: $\frac{\cos(x)}{x}$.

More generally, a function which is composed of continuous functions and evaluates to $\frac{a}{0}$ at a point where $a \neq 0$ must have an infinite discontinuity there.