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?
Best Answer
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.
Moreover L'Hospital's Rule should not be used instantly. It should be used only when other simpler techniques (algebra of limits, Squeeze theorem) fail. And even when you really need to apply this rule, it is better to simplify the expression using algebra of limits and usual algebraic manipulation. Jumping to L'Hospital's Rule for any and every limit problem is a bad bad bad idea.