.

Other singularities require more analysis to diagnose. Consider the functions $\frac{\sin x}{x}$, $\frac{\sin x}{\vert x\vert}$ and $\frac{\sin x}{1 - \cos x}$ at the point $x = 0$. All three functions evaluate to $\frac {0}{0}$ at that point, but have different kinds of singularities. The first has a removable discontinuity, the second has a finite discontinuity and the third has an infinite discontinuity. See Figure 13.5.

Figure: The functions $\frac{\sin x}{x}$, $\frac{\sin x}{\vert x\vert}$ and $\frac{\sin x}{1 - \cos x}$.

An expression that evaluates (for a particular value of the independent variable) to $\frac {0}{0}$, $\frac {\infty }{\infty }$, $0 \cdot \infty$, $\infty -\infty$, $1^\infty$, $0^0$ or $\infty ^0$ is called an indeterminate. A function $h(x)$ which takes an indeterminate form at $x = a$ is not defined for $x = a$ by the given analytical expression. For example, suppose we have

$$y = \frac{f(x)}{g(x)},$$

where for some value of the variable, as

$$f(a) = 0,\ \ \ \ g(a) = 0.$$

For this value of $x$ our function is not defined and we may therefore assign to it any value we please. It is evident from what has gone before (Case II, [§3.6]) that it is desirable to assign to the function a value that will make it continuous when $x = a$ whenever it is possible to do so.