Why Can L'Hôpital's Rule Not be Applied to the Sum or Difference of Limits? Consider $$\lim_{x\to\infty}\frac{f(x)}{g(x)} + \lim_{x\to\infty}\frac{h(x)}{i(x)}$$
I was told that I cannot apply L'Hôpital's rule to each individual limit and then join the limits as
$$\lim_{x\to\infty}\frac{f(x)}{g(x)} +\frac{h(x)}{i(x)}$$
Why is this incorrect?
 A: The following identity
$$\lim_{x\to \infty}\left(\frac{f(x)}{g(x)} +\frac{h(x)}{i(x)}\right)=\lim_{x\to \infty}\frac{f(x)}{g(x)} + \lim_{x\to \infty}\frac{h(x)}{i(x)}$$
doesn't hold in general and to solve the LHS limit by l'Hopital, if necessary, we need to put it in the form
$$\lim_{x\to \infty}\left(\frac{f(x)}{g(x)} +\frac{h(x)}{i(x)}\right)=\lim_{x\to \infty}\frac{f(x)i(x)+h(x)g(x)}{g(x)i(x)} $$
the reason is that the case you are referring to is not among the cases considered by l'Hopital theorem.
A: 
I was told that I cannot apply L'Hôpital's rule to each individual limit and then join the limits as ...

This has nothing to do with the L'Hôpital's rule itself.
The rule that you cannot use is:

$\lim\limits_{x\to...}f(x)+\lim\limits_{x\to...}g(x)=\lim\limits_{x\to...}(f(x)+g(x))$

And you can see from JDMan4444's answer that there are situations where this rule does not work.
However, if you are sure that $\lim\limits_{x\to...}f(x)$ and $\lim\limits_{x\to...}g(x)$ exist (and are not $\pm\infty$), you can apply that rule. (And of course you can apply L'Hôpital's rule to the sum in this case.)
This is important because in some cases it is possible to prove that both limits exist but calculating the limits directly is very difficult or even impossible. In such cases calculating the limit of the sum may be easier.
A: Consider the following example:
$$
\lim_{x\rightarrow\infty}\frac{x^2}{x}+\lim_{x\rightarrow\infty}\frac{-x^2}{x} = \infty - \infty \quad \text{(which is undefined)}
$$
$$
\lim_{x\rightarrow\infty}\frac{x^2}{x}+\frac{-x^2}{x} = 0
$$
A: It seems to me you got some bad, or incomplete, advice. It's not incorrect if both of those limits exist and are finite. This doesn't have much to do with L'Hopital. The $\infty - \infty$ or $-\infty + \infty$ cases are problematic whether you're contemplating L'Hopital or not.
