Confused about this limit If $\lim_{x \to \infty} (-1)^x$ is undefined ...
Why is $\lim_{x \to \infty} (-1/4)^x$ zero?
Couldn't you take out the negative to make it $\lim_{x \to \infty} (-1)^x$ * $\lim_{x \to \infty} (-1/4)^x$ which would make it undefined?  
Does undefined * 0 = undefined or 0?

You have the following: let $x\in\mathbb{N}$ then the sequence $a_x$ 
$$
a_x:=(-1)^x 
$$
can only take two possible values, $a_{2x}=1, a_{2x+1}=-1$ and therefore the limit does not exist.
If you now look to the sequence $b_x$ with
$$
b_x:=(-\frac14)^x 
$$
then you can make the following inequality work
$$
-(\frac14)^x\leq (-\frac14)^x\leq(\frac14)^x
$$
and $-(\frac14)^x,  (\frac14)^x$ both tend to zero as $x$ goes to  infinity. and therefore also $b_x=(-\frac14)^x $ goes to zero.
Now why can you not take out the $(-1)^x$ out of $b_x$? This is because the limit is not defined. if you want to 
$$
\lim_{x\to\infty}a_xb_x=\lim_{x\to\infty}a_x\lim_{x\to\infty}b_x
$$
both limits on the RHS have to exist.
