#### Answer

$[-4,\ 0) \cup (0, 1]$

#### Work Step by Step

We are given:
$f(x)=\displaystyle \sqrt{x+4}-\frac{\sqrt{1-x}}{x}$
The domain of $\sqrt{x+4}$ is: $[-4,\ \infty$) (since $x$ must be greater or equal to -4 to avoid a negative root).
The domain of $\displaystyle \frac{\sqrt{1-x}}{x}$ is: $(-\infty,\ 0)\cup (0, 1]$ (since $x$ must be less than or equal to 1 to avoid a negative root and not 0).
Therefore, the domain of $f(x)$ is the intersection of the two domains above: $[-4,\ 0) \cup (0, 1]$.