Solution: (1+sqrt(5))/2 .  

The concept of approaching a limit is usually introduced in calculus, but we can investigate it here. Denote the nth Fibonacci number as F_n. If the sequence of Fibonacci ratios converges to a fixed limit a, as suggested, then for large values of n, F_n+1 ≈ a · F_n and F_n+2 ≈ a² · F_n. 

Since each term is equal to the sum of the previous two, this gives F_n+2 = F_n+1 + F_n which implies a² · F_n = a · F_n + F_n, which simplifies to a² – a – 1 = 0. Applying the quadratic formula to this equation yields:

Only the positive result is reasonable in this situation, so the Golden Ratio is approximately 1.618.

Of course, the equations used above are only approximations, and we have assumed that a limit exists in the first place. However, if a limit does indeed exist, then it must equal the given value.