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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

05_limits_g_approx-91.gif

exists or is infinite, then

05_limits_g_approx-92.gif

(See Figure 5.2.2.)

05_limits_g_approx-94.gif

Figure 5.2.2: L'Hospital's Rule

Usually the limit will be given by

05_limits_g_approx-93.gif

and in this case the proof is very simple.

 

PROOF IN THE CASE

05_limits_g_approx-96.gif

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

05_limits_g_approx-97.gif

Taking standard parts we get

05_limits_g_approx-98.gif

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).

05_limits_g_approx-99.gif

Figure 5.2.3

The equation

05_limits_g_approx-100.gif

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

05_limits_g_approx-101.gif

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