Date: 12/23/98 at 11:43:47
From: Doctor Rob
Subject: Re: Proof of L'hopital's rulehttp://archives.math.utk.edu/visual.calculus/3/index.htmlOriginally posted by KL Kam
How to prove it? I've watched a tv program before and I vaguely remember the proof consists of 4 parts, which includes the mean value theorem, Cauchy's mean value theorem and the other 2 theorems. I haven't learned mean value theorem at that time and I didn't understand it, perhaps I can now.
L'Hospital's Rule is a mathematical theorem that allows us to evaluate limits of indeterminate forms by taking the derivative of the numerator and denominator and then evaluating the limit again.
The Mean Value Theorem is a fundamental theorem in calculus that states that if a function is continuous on a closed interval and differentiable on the open interval, then there exists a point in the open interval where the slope of the tangent line is equal to the slope of the secant line connecting the endpoints of the interval.
To prove L'Hospital's Rule with Mean Value Theorem, we first apply the Mean Value Theorem to the numerator and denominator of the original function. This will give us a new function that has the same limit as the original function. Then, by taking the derivative of this new function and evaluating the limit again, we can use L'Hospital's Rule to solve the limit.
The conditions for using L'Hospital's Rule with Mean Value Theorem are that the original function must be in an indeterminate form, the derivative of the numerator and denominator must exist and be continuous on the interval, and the limit of the derivative of the function must exist.
Proving L'Hospital's Rule with Mean Value Theorem is important because it provides a rigorous mathematical proof for a commonly used rule in calculus. It helps us understand the underlying principles and conditions for using L'Hospital's Rule, and allows us to apply it confidently in various mathematical problems.