342 Chapter 6 • Differentiable Functions
We can often evaluate an indeterminate form (1) by algebraically trans-forming ~~:~ into a form that is not indeterminate, and then taking the limit.
We did that when finding the derivative of a function using the definition. For
example,
(^1) lm. (x+h)2-x2 = 1. im (2 x - h) =^2 x.
h-o h h-o
L'H6pital's rule gives us a fresh approach to finding such limits. As you
probably remember from your elementary calculus course, L'H6pital's rule tells
us that under certain circumstances,
lim j(x) = lim f'(x).
x-a g(x) x-a g'(x)
. x^2 - 1. 2x
For example, bm --= bm - = 2.
x-1 X - 1 x-1 1
You will also remember that this rule is subject to certain limitations,
and care must be taken not to use it when it does not apply. For example,
x^2 2x
lim --=!= lim -.
x-l X + 1 x-1 1
L 'H6pital's rule is derived from a form of the mean value theorem, which
is why the topic is located here. We begin by proving this theorem.
Theorem 6.6.2 (Cauchy's Mean Value Theorem): Suppose f, g are con-
tinuous on [a, b], and differentiable on (a, b), where a< b. Then, :Jc E (a, b) 3
g'(c)[f(b) - f(a)] = J'(c)[g(b) - g(a)].Proof. Suppose f, g are continuous on [a, b], and differentiable on (a, b),
where a < b. As in the proof of the mean value theorem, we define a new
function h on [a , b] by
h(x) = f(x)[g(b) - g(a)] - g(x)[f(b) - f(a)].
Then his continuous on [a, b] and differentiable on (a, b) since f and g have
these properties. Moreover,
and
h(a) = f(a)[g(b) - g(a)] - g(a)[f(b) - f(a)]
= f(a)g(b) - g(a)j(b)h(b) = f(b)[g(b) - g(a)] - g(b)[f(b) - f(a)]
= g(b)f(a) - f(b)g(a).