What is values of exponential in complex plane I have a doubt about value of $e^{z}$ at $\infty$ in one of my book they are mentioning that as $\lim_{z \to \infty} e^z  \to \infty $
But in another book they are saying it doesn't exist.I am confused now 
As we can see $e^{z}$ is entire function then as $z \to \infty $ then $e^{z}$ must go to $\infty$
Please help.
 A: $|e^{i2\pi n}|=1$  for all $n$  and $|i2\pi n| \to \infty$. So it is not true that $|e^{z}| \to \infty$ as $|z| \to \infty$. Of course the limit is $\infty$ when you take limit through $\{1,2,..\}$ so the limit of $e^{z}$ as $|z| \to \infty$ does not exist. 
A: $e^z$ has an essential singularity at $z=\infty$ so we would not say $\lim\limits_{z\to\infty}e^z=\infty$, but instead we say the limit does not exist.
A: In real analysis,
$$
\lim\limits_{x\to\infty}e^x=\infty
$$
because the limit is taken along the positive real axis. Similarly,
$$
\lim\limits_{x\to-\infty}e^x=0
$$
because the limit is taken along the negative real axis.
However, in complex analysis, $\infty$ is taken to be the point added to make the one-point compactification of the complex plane (the Riemann Sphere). $e^z$ takes on all non-zero complex values in any neighborhood of $\infty$. Therefore, $\lim\limits_{z\to\infty}e^z$ does not exist.
A: I think $e^x$ goes to infinity, while $e^z$ or $(e^i)^x$ spins around the unit circle at the rate of the distance around a unit circle.
The letter 'z' could mean Real Integer or Complex number:
z = integers = ( -inf, ..., -2,-1,0,1,2....inf)
z = complex = x + iy
Which are you referring to?
A: Unless you are working on extended complex plane i.e. $\mathbb C\cup$ {$\infty$}, you can use the substitution $z=\frac{1}{w}$ and investigate the behaviour at $w=0$.
$e^{1/w}$ has essential singularity at $w=0$ and Big Picard's theorem says that it will take all values on $\mathbb C$ in the neighborhood of $w=0$ with atmost one exception.
