Let $f(x) = \left\lceil\dfrac{1}{x+2}\right\rceil$ for $x > -2$, and $f(x) = \left\lfloor\dfrac{1}{x+2}\right\rfloor$ for $x < -2$. ($f(x)$ is not defined at $x = -2$.) Which integer is not in the range of $f(x)$?
For $x > -2$, $\dfrac{1}{x+2}$ takes on all positive values. Thus, $f(x)$ takes on all positive integers for $x > -2$.

For $x < -2$, $\dfrac{1}{x+2}$ 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 $\boxed{0}$.