Proving Limits with Le'Hopital's Rule

[talolard]

Homework Statement

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 $$lim_{x-> \infty}f(x)= \frac{L}{a}$$

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 doesn't 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

Homework Statement

Homework Equations

The Attempt at a Solution

Homework Statement

Homework Equations

The Attempt at a Solution

Homework Statement

Homework Equations

The Attempt at a Solution

Homework Statement

Homework Equations

The Attempt at a Solution

Homework Statement

Homework Equations

The Attempt at a Solution

 

[vela]

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}.

 

[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