Reference: Thomas, Finney, 9th edition, Calculus and Analytic Geometry.
Continuing our previous discussion of “theoretical” calculus or “rigorous” calculus, I am reproducing below the proof of the finite limit case of the stronger form of l’Hopital’s Rule :
L’Hopital’s Rule (Stronger Form):
Suppose that
and that the functions f and g are both differentiable on an open interval that contains the point
. Suppose also that
at every point in
except possibly at
. Then,
….call this equation I,
provided the limit on the right exists.
The proof of the stronger form of l’Hopital’s Rule is based on Cauchy’s Mean Value Theorem, a mean value theorem that involves two functions instead of one. We prove Cauchy’s theorem first and then show how it leads to l’Hopital’s Rule.
Cauchy’s Mean Value Theorem:
Suppose that the functions f and g are continuous on and differentiable throughout
and suppose also that
throughout
. Then there exists a number c in
at which
…call this II.
The ordinary Mean Value Theorem is the case where .
Proof of Cauchy’s Mean Value Theorem:
We apply the Mean Value Theorem twice. First we use it to show that . For if
did equal to
, then the Mean Value Theorem would give:
for some c between a and b. This cannot happen because
in
.
We next apply the Mean Value Theorem to the function:
.
This function is continuous and differentiable where f and g are, and . Therefore, there is a number c between a and b for which
. In terms of f and g, this says:
, or
, which is II above. QED.
Proof of the Stronger Form of l’Hopital’s Rule:
We first prove I for the case . The method needs no change to apply to
, and the combination of those two cases establishes the result.
Suppose that x lies to the right of . Then,
and we can apply the Cauchy’s Mean Value Theorem to the closed interval from
to x. This produces a number c between
and x such that
.
But, so that
.
As x approaches , c approaches
because it lies between x and
. Therefore,
.
This establishes l’Hopital’s Rule for the case where x approaches from above. The case where x approaches
from below is proved by applying Cauchy’s Mean Value Theorem to the closed interval
, where
. QED.