**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.**

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