When to Use L'Hôpital's Rule

The textbook explanation shows that L'Hôpital's rule can be used on a rational function ${f(x)}\over {g(x)}$ if it is continuous, and $\lim_{x \to c}f(x) = \lim_{x \to c}g(x) = 0$ or $\pm \infty$, and $g'(x) \neq 0$.

The textbook doesn't explain, however, what steps are necessary to take before I dive blindly into applying L'Hôpital's rule e.g. is there a more elegant way to solve the limit?

Usually I would assume that ${f(x)}\over {g(x)}$ is in an irreducible form and the limit still evaluates to $0 \over 0$ or $\infty \over \infty$. If I can't find any terms to cancel out, then I would proceed to use L'Hôpital's rule ...

I guess what I'm asking is what is the best approach to any limit problem? How do I know when I should break out L'Hôpital's rule?

• Uhh, have you checked the provided proof? – Benicio Oct 27 '15 at 7:04
• L'Hôpital's rule saves time and helps in easily evaluating imits. But if you want elegant proofs, then this will be your last choice and the conventional method will suit you better. – SchrodingersCat Oct 27 '15 at 7:04
• Not more elegant, but easier to handle (esp. if several rounds of l'Hôpita are needed) is to use Taylor expansions ... – Hagen von Eitzen Oct 27 '15 at 7:13
• The post seems to be specifically about rational functions. If top and bottom have limit $0$ at $c$, then $x-c$ divides both, so in principle one can do the division. It seems harmless to use L'Hospital to bypass division. – André Nicolas Oct 27 '15 at 7:14
• I tell students that l'Hôpital's theorem is something like the Paris gun: a very powerful weapon, but that can do damage for no good. – egreg Oct 27 '15 at 9:39

Let's suppose we are dealing with limits of the form $x \to a$. Then we have two cases where L'Hospital's Rule is applicable.
Theorem 1: Let $f, g$ be function differentiable in a deleted neighborhood of $a$. If $f(x) \to 0, g(x) \to 0$ as $x \to a$ and $f'(x)/g'(x) \to L$ as $x \to a$ then $f(x)/g(x) \to L$ as $x \to a$.
This is the typical case of indeterminate form $0/0$. Note that the result is valid even if $L$ above is infinite.
Theorem 2: Let $f, g$ be function differentiable in a deleted neighborhood of $a$. If $|g(x)| \to \infty$ as $x \to a$ and $f'(x)/g'(x) \to L$ as $x \to a$ then $f(x)/g(x) \to L$ as $x \to a$.
This is the case where denominator $g(x)$ tends to infinity. There is no need to check the behavior of numerator here.