What is the $\lim_{z\rightarrow{\infty}}e^{-(z+i)^2}$?

Question

I'm struggling to find the following $complex$ limit (or show that it DNE): $\lim_{z\rightarrow{\infty}}e^{-(z+i)^2}$.

I know that we are actually looking at $\lim_{|z|\rightarrow{\infty}}e^{-(z+i)^2}$ where $|z|$ is the modulus of $z \in \mathbb{C}$.

For example $\lim_{z\rightarrow{\infty}}e^{-z}$ does not exist because $\lim_{x\rightarrow{\infty}}e^{-x} \neq \lim_{x\rightarrow{-\infty}}e^{-x}$ (where $z$ is taken $z=x+0i)$.

I tried to do a similar thing in this problem but it doesn't work. Anny ideas here, does the limit exist and can we use a theorem or do we need to go back to the definition?

Answer 1

That limit does not exist, because if $z$ is of the form $-i+\lambda$, with $\lambda\in\Bbb R$, then the limit is $0$. But if $z$ is of the form $-i+\lambda i$, with $\lambda\in\Bbb R$, then the limit is $\infty$.

Answer 2

If $z = ix$ for real $x$, as $x \to \infty$, $(z+i)^2 =(ix+i)^2 =-(x+1)^2 $ so $e^{-(z+i)^2} \to \infty$.

If $z = x-i$ for real $x$, as $x \to \infty$, $(z+i)^2 =(x-i+i)^2 =x^2 $ so $e^{-(z+i)^2} \to 0$.

Therefore the limit does not exist.