## L'Hospitals Rule for Division by Zero

L'HOSPITAL'S RULE FOR 0/0

Suppose that in some deleted neighborhood of a real number c, f'(x) and g'(x) exist and g'(x) ≠ 0. Assume that

limx→c f(x) = 0, limx→c g(x) = 0.

If

exists or is infinite, then

(See Figure 5.2.2.)

Figure 5.2.2: L'Hospital's Rule

Usually the limit will be given by

and in this case the proof is very simple.

PROOF IN THE CASE

Let Δx be a nonzero infinitesimal. Then f(c) = 0, g(c) = 0, and

Taking standard parts we get

Intuitively, for x ≈ c the graphs of f(x) and g(x) are almost straight lines of slopes f'(c), g'(c) passing through zero, so the graph of f(x)/g(x) is almost the horizontal line through f'(c)/g'(c) (Figure 5.2.3).

Figure 5.2.3

The equation

is not always true. For example, g'(c) might be zero or undefined.

is sometimes another limit of type 0/0, that is,

limx→c f'(x) = 0 and limx→c g'(x) = 0.

When this happens, l'Hospital's Rule can often be reapplied to limx→c f'(x)/g'(x). The proof of l'Hospital's Rule in general is fairly long and uses the Mean Value Theorem. It will not be given here.

Here are some examples showing how the rule can be applied.

 Example 1
 Example 2
 Example 3
 Example 4
 Example 5

Last Update: 2010-11-25