L'Hôpital's rule only applicable if right-hand limit exist? In a source I have been reading, this statement was made regarding L'Hôpital's rule:



Why is it the case that L'Hôpital's rule is applicable only if the right-hand limit exists?  Why not the left-hand? Why not both?  I have read other sources on L'Hôpital's rule that do not mention it and would like clarification.
Here is the text:

L'Hôpital's Rule: Suppose that $f$ and $g$ are differentiable functions, and $f(a)=g(a)=0$, and suppose that $g'(x)$ is nonzero in a neighborhood of $a$ (except maybe at $a$ itself).  Then
  $$\lim\limits_{x \to a} \ \frac{f(x)}{g(x)}=\lim\limits_{x \to a} \ \frac{f'(x)}{g'(x)}$$
  if the limit on the right-hand side exists.


My pocket example of such a limit is
$$ \lim_{x \to \infty} \frac{x + \sin x}{x + \cos x}.$$
This limit is very clearly $1$. But an application of l'Hopital's rule would lead to the consideration of
$$ \lim_{x \to \infty} \frac{1 + \cos x}{1 - \sin x},$$
which doesn't exist! Thus existence of the left hand limit does not guarantee the existence of the right hand limit.
