Showing that $e^z$, $\sin(z)$, $\cos(z)$ have essential singularities at $\infty$ 
Problem: Show that $e^z$, $\sin(z)$, $\cos(z)$ have essential singularities at $\infty$.

Attempt for $e^z$:


*

*We have that $e^z$ has an essential singularity at $\infty$ iff $e^{1/z}$ has an essential singularity at $0$ iff
$$
0 \ne \lim_{z \to 0} |z|^{\alpha} |e^{1/z}| \ne \infty
$$
for all values of $\alpha \in \mathbb{R}$.
My intuitions must be off here, because it seems to me like $|e^{1/z}|$ tends to infinity much quicker than $|z|^\alpha$ tends to zero.  That is, it seems to me that indeed
$$\lim_{z \to 0} |z|^{\alpha} |e^{1/z}| = \infty$$
which evidently is not the case.
 A: $$
e^z=e^{x+iy}\quad \Big( \text{where }x,y\in\mathbb R \Big) = e^x (\cos y+i\sin y).
$$
So think about what happens as $y\to\infty$ with $x$ fixed.  No limit in $\mathbb C\cup\{\infty\}$ is approached.
A: $f(z)$ has an essential singularity at $\infty_{\mathbb C}$ iff $f(1/z)$ has an essential singularity in $0$.
Now the Taylor series of $e^z$ (around $z=0$, but it holds on all $\mathbb C$) is
$$
e^z=\sum_{n=0}^{+\infty}\frac{z^n}{n!}
$$
Hence
$$
e^{\frac1{z}}=\sum_{n=0}^{+\infty}\frac1{z^n n!}
$$
that is the Laurent series for $e^{1/z}$ around $z=0$.
So $c_{-n}\neq0$ for (almost) all $n\in\mathbb N$, that is the definition of essential singularity.
Proceed similarly for $\sin$ and $\cos$.
A: Since we have that
$$
\lim_{z \to 0+} \left|e^{1/z}\right| = \infty
$$
$$
\lim_{z \to 0-} \left|e^{1/z}\right| = 0
$$
it follows that
$$
0 \ne \lim_{z \to 0} |z|^{\alpha} |e^{1/z}| \ne \infty
$$
so that $e^z$ has an essential singularity at $\infty$ by definition.
Furthermore, the definitional dependencies of $\sin(z)$ and $\cos(z)$ on the exponential function means that
$$\cos(z) = {e^{iz} + e^{-iz} \over 2}$$
$$\sin(z) = {e^{iz} - e^{-iz} \over 2i}$$
have essential singularities at $\infty$ if and only if $\exp(z)$ has an essential singularity at $\infty$.  Therefore, both $\cos(z)$ and $\sin(z)$ have essential singularities at $\infty$ as well. This completes the proof.
