Understanding the L’Hôpital Rule

If you are struggling trying to understand the L’Hôpital Rule, also called the L’Hospital Rule (with the ‘s’ silent), this post if for you. I assume you know basic calculus—at least what is a derivative.

Let f(x) and g(x) be two different functions of x, where f’(x) and g’(x) are their corresponding first derivatives.

Assume that we want to evaluate the f(x)/g(x) ratio when x = c, where c can be any real number. If direct substitution of c for x yields an indeterminate form, this obstacle is circumvented by evaluating the f’(x)/g’(x) ratio when x = c.

This is the so-called L’Hôpital Rule.

If the result still is an indeterminate form, we keep applying the L’Hôpital Rule to the result until said forms no longer are. To illustrate, consider the following example, where f(x) = (1 – cos x) and g(x) = x^2. If we evaluate the ratio f(x)/g(x) when x = c = 0,

f(x)/g(x) = 0/0

and this result asks for applying the rule. It can be shown that f’(x)/g’(x) when x = c = 0 yields

f’(x)/g’(x) =  (sin x)/(2x) = 0/0

So, we keep applying the rule. Then, f’’(x)/g’’(x) when x = c = 0 yields

f’’(x)/g’’(x) = (cos x)/2 = ½.

In an upcoming post, we will explain how to use the L’Hôpital Rule to derive the geometric mean from the power mean.