Reducing number of limit statements I am terrified of making a mistake of forgetting to write lim on every side of a limit expression chain. I always end up forgetting one, and mess up readability of my tests by arrows and statements on top of page.
Is there a way to not write lim all over the page, or write it less often?
for example
$$\begin{align*}\lim _{h\to0}\frac{(x+h)^2 - x^2}h &= \lim_{h\to0}\frac{(x^2+2xh+h^2)-x^2}h = \lim_{h\to0} \frac{2xh+h^2}h \\&= \lim_ {h\to0}\frac{h(2x+h)}h = \lim_{ h\to0}2x+h = 2x\end{align*}$$
the above had so many $\lim$s... it's aggrevating...
Is there a way to make it obvious that this block's h is a limit, some sort of distributive limit that applies to each individual part of the equality chain?
Thanks!

If the most complicate part in the proof is writing the lim you are a blessed person. 
I wouldn't recommend it but you can write all steps where you don't take a limit.
That would be like, for $h\neq 0$ we have:
$$\frac{(x+h)^2-x^2}{h} = \frac{x^2 +2hx+h^2 -x^2}{h^2}= \frac{2hx+h^2}{h}=\frac{h}{h} \cdot (2x+h)=2x+h$$ 
Hence 
$$\lim_{h\to 0} \frac{(x+h)^2-x^2}{h}=\lim_{h\to 0} 2x+h = 2x $$
You should always think about what is more legible and what makes more sense in the proof. 
