Is
\[f(x) = \frac{1}{x + 2} - \frac{1}{x - 2}\]an even function, odd function, or neither?

Enter "odd", "even", or "neither".
Since
\begin{align*}
f(-x) &= \frac{1}{-x + 2} - \frac{1}{-x - 2} \\
&= -\frac{1}{x - 2} + \frac{1}{x + 2} \\
&= f(x),
\end{align*}$f(x)$ is an $\boxed{\text{even}}$ function.

Note that
\[f(x) = \frac{1}{x + 2} - \frac{1}{x - 2} = \frac{(x - 2) - (x + 2)}{x^2 - 4} = -\frac{4}{x^2 - 4}.\]In this form, it is clear that $f(x)$ is even.