Large deviation principle and Fibonacci sequence

[dyh]

Homework Statement

A>0, B can be any number

Homework Equations

To show that

lim(n->∞) (1/n)log[A((1+sqrt(5))/2)^n +B((1-sqrt(5))/2)^n] = (1+sqrt(5))/2

The Attempt at a Solution

I used Lhopital's Rule to solve this and got
log((1+sqrt(5))/2)

So, I don't know what is wrong.
If you guys know how to prove this one, please let me know.
Thanks a lot

 

[HallsofIvy]

If you used L'Hopital's principle, how do you still have a logarithm?

 

[dyh]

Because if I differentiate A*(1+sqrt(5))^n with respect to n

then I would get

A*log(1+sqrt(5))*(1+sqrt(5))^n