Why is function F used in the proof of l'hopitals rule?

[georg gill]

This is the proof for l'hopitals rule in my book:

http://bildr.no/view/953799

http://bildr.no/view/953800

http://bildr.no/view/999144

I don't get why they use the function F to make the proof? It is the saying from it with cauchy mean value theorem that makes the start for the proof but could they not manipulate any function to make this proof work, I mean it looks like they just made up F? And at the end of the proof I don't get why f(a)=g(a)=0

Hope someone could help me out:)

 

[Devil Doc]

This is my first time on here. Did you just post this question today?

 

[georg gill]

Yes I did. Just can't seem to get it right. I am also a relatively newbie here

 

[Devil Doc]

Same here. That's why I asked the question. My professor just covered L'Hopital's Rule today. This is what I got from my notes. First, we're going to use L'H's Rule to help evaluate indeterminate forms such as 0/0 and infinity/infinity. Note: If we don't have these forms, we can just use an alternative.

1. L'H's Rule is intended for Quotients.
2. 0/0 or infinity/infinity
3. lim f'(x)/g'(x) exists

If this is true, then lim as x approaches a f(x)/g(x) = lim as x approaches a f'(x)/g'(x)

Maybe we can figure this thing out together.

 

[Devil Doc]

Note: I haven't actually taken the time to go through my book's proof as of yet. But I have started my homework.

 

[georg gill]

Sure:) I'll try to find another proof online because I am stuck in the one in my book.

 

[georg gill]

Found an old post about the same. I tried to ask there as well:

https://www.physicsforums.com/showthread.php?p=3559086#post3559086

 

[Devil Doc]

You can try this link:

http://math.chapman.edu/~jipsen/mathposters/L'Hospital's Rule.pdf

I know why. f(a) = g(a) = 0 is just another way of saying f(x) = 0 = g(x). In other words, we want to use L'H's rule to help evaluate indeterminate forms like 0/0 and infinity/infinity. So would you agree that f(a)/g(a) = 0/0 because f(a) = 0 and g(a) = 0, right? If this is indeed the case, then we'll use L'Hopital's Rule.

One more thing, the mean theorem states that there exists a f(c) between a and b, correct? And it must be differentiable, correct? Then if that is the case, we can use L'Hopital's Rule to help us evaluate indeterminate forms such as 0/0 and infinity/infinity.

The same is true for infinity/infinity. Go back to what I said about the function equaling the derivative. If those 3 points are true, then we can find the derivative of the function to help us evaluate the problem.

So as you can see, the a's and x's are just different symbols, but they mean the same thing.

Does this help? I am trying to repeat the important parts over and over again for our benefit.

 

[Devil Doc]

I like this site. This page also has the proof. And it's very understandable, at least I think.

http://www.math.hmc.edu/calculus/tutorials/lhopital/

 

[miglo]

georg gill i have the same calculus book as you i think, i checked out the proof and your question concerning the function F(x), I am not 100% sure about this but it looks to me that the line in figure 4.42 in your book, the one extending from (g(b),f(b)) to (g(a),f(a)), the equation for that line is F(x). As for the rest of the proof for Cauchy's mean value theorem and L'hopital's rule I am not so sure of, so hopefully someone else can help us understand the proof.

 

[georg gill]

I am trying to get one or anoher proof for this and the thing i might get is why they can use F because I got help from another forum (norwegian as I am from Norway so I guess showing you that won't be any use)

Say you have the function:

f(x)=x-2

We want to rewrite to another function

F:

F(x)=g(x)-x-2

everytime F=0 then we have a new saying for x-2 that is g.

and this is what they do in the start of the proof i guess. So I only wonder from the proof that Devil Doc gave why they can assume that f(a)=g(a)=0

 

[georg gill]

You can try this link:

http://math.chapman.edu/~jipsen/mathposters/L'Hospital's Rule.pdf

I know why. f(a) = g(a) = 0 is just another way of saying f(x) = 0 = g(x). In other words, we want to use L'H's rule to help evaluate indeterminate forms like 0/0 and infinity/infinity. So would you agree that f(a)/g(a) = 0/0 because f(a) = 0 and g(a) = 0, right? If this is indeed the case, then we'll use L'Hopital's Rule.

We know that in x=a we have a saying which is 0 over 0 for f(a) over f(g). That is why we can delete this part in the saying and get L'hopital's formula in my book. That helped. Thanks Devil Doc

 

[Devil Doc]

You're welcome, Georg. I'm glad it helped. It helped me to explain it. If you have any homework problems that your having trouble with, feel free to post them. I wouldn't mind taking a crack at them. I will do the same if your interested.

- Doc

 

[georg gill]

Sure I can do my best to help as well! I am taking elementary math for college now. And great to have some place to ask:)