1549901369-Elements_of_Real_Analysis__Denlinger_

6.6 *L'Hopital's Rule 343

Thus h(a) = h(b), so h satisfies all the hypotheses of Rolle's theorem.

Hence, :3 c E (a, b) 3 h' ( c) = 0. By definition of h,

h'(x) = J'(x)[g(b) - g(a)] - g'(x)[f(b) - f(a)].

Since h'(c) = 0, we have g'(c)[f(b) - f(a)] = f'(c)[g(b) - g(a)]. •

Geometric interpretation of the Cauchy mean value theorem: In
elementary calculus, we studied curves given in parametric form: x = f (t), y =
g(t), a::::; t::::; b. We learned t here that if f and g are differentiable on (a, b), then
for any t E (a, b) where f' ( t) =/:-0, the slope of t he curve at the point (f ( t), g( t))
g'(t)
ism= f'(t). Suppose that Vx E (a, b), f'(x) =/:-0. Then the conclusion of the

Cauchy mean value theorem can be written

g'(c) g(b) - g(a)

:3 c E (a, b) 3 f'(c) = f(b) - f(a).

Note that when f'(x) =/:- 0 on (a, b), Rolle's t heorem guarantees that f(a) =f-
f(b). The Cauchy mean value theorem thus says that under the above condi-
tions, there is some value c E (a, b) for which the slope of the line tangent to
the curve at (f(c), g(c)) is equal to the slope of the secant line through the
endpoints of the curve, (f(a),g(a)) and (f(b),g(b)). (See Figure 6.8.)

Figure 6.8

L'Hopital's rule covers a multitude of cases. We cannot give a single proof
appropriate for this course that covers all these cases; we must look at individual
cases separately. Here is a comprehensive statement that incorporates all cases.