Finding a limit help.

1. Sep 18, 2012

Eats Dirt

1. The problem statement, all variables and given/known data

Finding the limit of (1/n)^(1/ln(n))

as n-> infinity

2. Relevant equations

Log rules

3. The attempt at a solution

so I take the ln of both sides and get

(1/ln(n))*ln(1/n) in an attempt to get it into the proper condition for l'hopitals rule.

1/ln(infinity) is zero but ln(1/n) is undefined and I have the same problem when trying to multiply by ((1/n)/(1/n)) because it is still an indeterminate form and I can not apply l'hopitals rule.

2. Sep 18, 2012

Staff: Mentor

Let y = (1/n)(1/ln(n))
Then ln y = 1/(ln(n)) * ln (1/n) = 1/(ln(n)) * (-ln(n)) = (-ln(n))/ln(n)

Now take the limit, noting that for all finite n, the right side above equals -1. Note also that you can switch the order of the lim operation and the ln operation under certain conditions.

Does that get you started?

3. Sep 18, 2012

Eats Dirt

thanks!

4. Sep 18, 2012

Eats Dirt

Hey thanks ya this does help a lot but I didn't even know about that rule for ln!!
so ln(1/n) = -ln(n) ?!

I wonder how many other rules like this there are that I don't even know about haha.

Thanks so much.

5. Sep 18, 2012

Staff: Mentor

Yes. It's a special case of ln(A/B) = ln(A) - ln(B), with A = 1.