# Le'Hopital Rule

1. May 18, 2010

### talolard

1. The problem statement, all variables and given/known data
Use the Le'Hopitals rule to prove the following. Let F be differentiable on (0, infinity) and a> 0. If
$$\lim_{x-> \infty} (af(x)+2x^{1/2}f'(x)) =L$$ then $$lim_{x-> \infty}f(x)= \frac{L}{a}$$

3. The attempt at a solution
It seems to me that what I need to prove is that
$$lim _{x-> \infty} 2x^{1/2}f'(x)) =0$$ I'm pretty lost as to how to go about it,
I've tried to define a function $$g(x) = \frac{f(x)}{sqrt(x)}$$ but that didn't help.
My problem are that:
1. I don't know if f' is differentiable or not.
2. if f converges to L/a then it doesnt satisfy the requirments of the Le'Hopital rule.
3. I'm inclined to say that if a function converges then it's derivative converges to 0. This seems intuitive but I'm not sure how to prove it.

Thanks for the help
Tal
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

Last edited: May 18, 2010
2. May 18, 2010

### vela

Staff Emeritus
You're probably not going to get any useful responses until you fix all the typos in your post so that it makes sense. In particular, exactly what has a limit of 0 as x goes to infinity?

Also, use a backslash before "lim" and "sqrt" in your LaTeX to get those elements to appear correctly, e.g. \lim_{x\rightarrow\infty} and \sqrt{x}.

3. May 18, 2010

### talolard

Hey Vela,
Thanks for pointing that out, i hope its clearer now. I added the slash before lim but it didnt change much.
Thanks
Tal