L Hospital's Rule - AP Calculus

In calculus, L Hospital's Rule uses derivatives to evaluate limits involving indeterminate forms. In the AP Calculus exam you will be required to be familiar with L Hospital's Rule and calculate various limits, some of which may result indeterminate forms.

An indeterminate form in this context is a fraction that simplifies to 0/0 or ∞/∞. Obviously, 0/∞ or ∞/0 do not present a problem since they will evaluate to 0 or ∞ respectively.

However, when the numerator of a fraction is zero, that fraction will be equal to zero unless the denominator is also zero. Similarly, when the denominator is zero, the fraction will evaluate to  infinity unless the numerator is also zero. Similar reasoning applies to ∞/∞.

So the question is whether the numerator or the denominator wins in these cases. On the other hand, fractions in which the numerator and the denominator are identical evaluate to 1, so is this the answer?

L'Hospital's rule (also written as L'Hôpital's rule) may help help shed some light on these indeterminate forms. You have probably seen the rule in action, but here I want to give you some intuition about why it works.

Let us say we have two continuous, differentiable functions, f(x), and g(x) and we are asked to find the limit as x tends to some number c of f(x)/g(x). Problems arise when f(c) = g(c) = 0. In this case,

Since  f(c) = g(c) = 0, we can do this without getting into trouble. Now let us divide both numerator and denominator by (x – c).

The limit as x tends to c of the numerator on the right side is just the derivative of f(x) evaluated at x = c , f '(c), while that of the denominator is g ' (c). See my earlier post on derivatives if this is unclear.

As long as the derivatives of f and g exist at x = c and g ' (c) ≠ 0, this proves L'Hospital's rule for the indeterminate form 0/0.

Proofs for the other indeterminate forms are a bit more complicated than you need to know for the AP Calculus exam, but I wanted to show you this so that you realize that it isn't something pulled out of thin air. In the next post we will look at some examples of how to use L'Hospital's rule.