How do I factor equations involving $e^x$?

How do I factor equations involving $e^x$?

I was reviewing some of my notes from Calculus 1 so that I can prepare for
Calculus 2 this fall, and I ran into one problem where I don't understand
how the factoring works.
$$\lim_{x\to\infty} \frac{e^{1/x}+e^{-1/x}}{e^{1/x}-e^{-1/x}}$$
Then substituting... $$u = \frac {1}{x}$$
I know what the answer to this question is, but this step in the solution
manual confused me:
$$\lim_{x\to\infty} \frac{e^u(1+e^{-2u})}{e^u(1-e^{-2u})}$$
My brain is still getting prepared to get back into doing math regularly
after the summer vacation, but I'm hitting a serious wall trying to
understand how this expression can be factored this way.
Any help in understanding this is greatly appreciated!