Calculate $\lim_{z\to \infty}\frac{\cos(i+z)-1}{(z+i)^4}$

Question

Calculate $\displaystyle\lim_{z\to \infty}\frac{\cos(i+z)-1}{(z+i)^4}$

I'm pretty sure it's equal to $0$, but I find it difficult to show it formally.

Answer 1

If $z\to\infty$, then also $z+i\to\infty$ and conversely. So your limit is the same as $$ \lim_{z\to\infty}\frac{\cos z-1}{z^4} $$ Now $$ \cos z=\frac{e^{iz}+e^{-iz}}{2} $$ For real $z$, the limit is $0$. For $z=it$, $t>0$, the limit is $$ \lim_{t\to\infty}\frac{e^{-t}+e^{t}}{t^4}=\infty $$