I am having some difficulty evaluating and visualizing limits of complex rational functions. For example, $$ \lim_{z \to z_0 } f(z),\ where\ Im(z_0)\ne 0,\ Re(z_0)\ne 0$$ $$f(z):=\frac{L(z)}{M(z)}, where \ M(z)\ne 0$$
I know that the general epsilon-delta definition of a limit is akin to its counterpart in real analysis, but I admit that I am having some difficulty trying to use it to evaluate limits of complex rational functions when $z_0$ has non-zero real and imaginary parts.
$$\forall \epsilon_{>0} \exists \delta_{>0} \ |f(z)-L|<\epsilon\ whenever\ 0<|z-z_0|<\delta$$
Would it help to mentally re-formalize the above epsilon-delta definition as follows to make the definition more concise?
$$\forall \epsilon_{>0}\ \exists \delta_{>0} |Re(f(z))-Re(L)|<Re(\epsilon)\ whenever\ 0<|Re(z)-Re(z_0)|<Re(\delta);$$ $$\forall \epsilon_{>0}\ \exists \delta_{>0} |Im(f(z))-Im(L)|<Im(\epsilon)\ whenever\ 0<|Im(z)-Im(z_0)|<Im(\delta);$$
You should be thinking about discs, not intervals: $f(z)$ tends to $L$ at $z_0$ iff for any positive radius $\epsilon$, there is a positive radius $\delta$, such that for all $z$ in the disc of radius $\delta$ around $z_0$, $f(z)$ is in the disc of radius $\epsilon$ around $L$. This means that $f(z)$ tends to $L$ as $z$ tends to $z_0$ from any direction with a rate of convergence that is within a constant factor of the rate of convergence from any other direction.