Mathematics GRE

Any hint on the limit question?

I know dorminant convergence can be applied perfectly here, but any other method that solve the limit directly?

Thanks!

What is dominant convergence?

I just evaluated the integral, considering that, since n grows to infinity, I can consider it as greater than 1 and forget about the possibility of having to integrate 1/x. YThen I evaluated from 1 to n, and they found the limit as x goes to infinity.

Int [x from 1 to n] x^-n = [n^(-n+1)]/(n+1) - 1/(-n+1) =
1/[n^(n-1) * (n+1)] - 1/(-n+1) = F(n)

As n grows to infinity, the limit of F(n) is 0.

What do you think?