I'm trying to solve a complex variable limit question. It proceeds as follow:
Definition: We say that $\lim_{z \to \infty} f(z) = \infty$ if for any $R>0$ there exits $S>0$ such that if $|z|>S$, then $|f(z)|>R$
Use the above definition to prove that $\lim_{z \to \infty} \frac{z^3}{z^2+4z} = \infty$
This is how I proceed with the question: Given any R we have
$|\frac{z^3}{z^2+4z}| >R $, then $|\frac{z^2}{z+4}| = |z||\frac{z}{1+\frac{4}{z}}| > R$. Now we know that since z tends to infinity $|z| > 1$ so $|1+\frac{4}{z}| < 4 ...$. However, that's all I got up-to. I was thinking of showing the the fraction $|\frac{z}{1+\frac{4}{z}}| < X$ then $S = \frac{R}{X}$ would meet the inquelity. Any help ? Many thanks in advance.