Cross multiplication in limits $\lim_{h \rightarrow 0}\frac{f(h)}{h}= 5$
After doing something like cross multiplication, we get
$\lim_{h \rightarrow 0}f(h)=\lim_{h \rightarrow 0}5h+gh$
where $g$ is a function of $h$ and $g$ tends to $0$ as $h$ tends to $0$. 
How?

This is simple device used commonly in many proofs. Just note that if $\lim_{h\to 0}f(h)/h=5$ then $\lim_{h\to 0}(f(h)/h)-5=0$ and if we set $g(h) =(f(h)/h)-5$ then we get $\lim_{h\to 0}g(h)=0$. Thus we have $f(h) =5h+hg(h)$ where $g$ is a function of $h$ such that $g(h) \to 0$ as $h\to 0$. Thus you can see that this is nothing but a simple application of algebra of limits. 
