jmc

For $x>-2$, $\frac{1}{x+2}$ takes on all positive values. Thus, $f(x)$ takes on all positive integers for $x>-2$.

For $x<-2$, $\frac{1}{x+2}$ takes on all negative values. Thus, $f(x)$ takes on all negative integers for $x<-2$.

So, the range of $f(x)$ is all integers except for $\boxed{0}$.