Solving Limit: n/log(n) as x Approaches 0

[Daveyboy]

Homework Statement

lim x->0 (n/log(n))[n^1/n-1]

The Attempt at a Solution

I've just been trying to move things around, and use L'Hospitals when appropriate. I haven't been able to see the trick.

Got any hints?

 

[Focus]

Try the algebra of limits

 

[yyat]

Do you mean lim n->infinity?

 

[Daveyboy]

woops, you as n goes to infinity.

Ive been using L'hospitals on the numerator after multiplying the n through and rewriting in exponential form.

d/dx:[x^1/x+1-x] = d/dx:[e^(1/x+1)ln(x)-x]= e^(1/x+1)ln(x)( -ln(x)/x²+(x+1)/x²) -1
and the bottom goes to 1/x

so the new limit after applying L'Hospital is
e^(1/x+1)ln(x)((x+1)-ln(x)/x) -x

which leaves me with a big question mark.

 

[Daveyboy]

also tried looking at it like this.

(n^1/n-1)/(log(x)/x)

using L'H on the top and bottom give:

x^1/x((1-ln(x)/x²)/(1-log(x))... I want to do something with that log