Proving Limit of ln(x)/x as x-->∞ is 0

[battousai]

Homework Statement

Prove that the limit of (ln x/x) as x approaches to infinity equals 0 starting with the definition of ln x as an integral. (How do you type a limit with LaTex?)

Homework Equations

\int \frac{1}{x}\,dx = ln lxl + C

The Attempt at a Solution

I don't even know how to begin the solution. Unfortunately we don't really do much proofs in math Calc BC class. I found this problem online and I thought it was interesting. Any tips to get me started would be appreciated.

 

[╔(σ_σ)╝]

Do you mean a rigorous proof ?

Anyway divide the integral by x and use L'hospitals.

 

[battousai]

Hmm I would think a rigorous proof. I did think of L'Hopital's, but I thought that was too easy.

It's just a problem I saw online so IDK exactly what kind of proof it's looking for.

 

[╔(σ_σ)╝]

You mean with epsilons and delta's ?

Well using the precise definition of a limit to do the proof is not too easy.
Write down your question precisely with epsilon and delta then see if you could make some approximations. You are likely going to run into some difficulties.

I think L'hostipals is the way to go.

 

[icystrike]

\frac{\int\frac{1}{x}dx}{x}
Now use L'Hopital Rule..
You will be \frac{1}{x}
Then you allow ur limit to tend to infinity.