For example, is the function $\frac{\sin x}x$ periodic? Using the the definition of periodic functions that $f(x)=f(x+p)$, the following must be true if the function is periodic:
$\frac{\sin x}x=\frac{\sin (x+p)}{x+p} = \frac{\sin (x)\cos (p)+\sin (p)\cos (x)}{x+p}$
This means that the following must be true for the equation to be true:
$\sin p = 0$
$\therefore p = 2\pi n$
$\cos p = 1$
$\therefore p = 2\pi n$
$p = 0$ (for the denominator to be equal)
Which gives a consistent result only if $p=0$. It can also be seen that the denominator is responsible only for changing amplitude and if the function is graphed it has a consistent period. Is the function periodic?