Well it isn’t really calculus as there’s no differentiation (I guess you could consider the last step where you take a limit to be calculus), but it’s a bit like differential equations. You can write down the recurrence relation:

  a_(n+2) = a_(n+1) + a_n

Observe that there is a linear solution space (I.e. if you add solutions point wise or multiply each value by the same scalar, you get solutions), and the values a_0 and a_1 are sufficient to determine the sequence. Now guess that a_n = k^n is a solution:

  k^2 = k + 1
  (k - 1/2)^2 = 5/4
  k = (1 +/- sqrt(5))/2
  k = φ or -1/φ, where φ is the golden ratio

Due to linearity, there are a family of solutions a_n = Rφ^n + S(-1/φ)^n for any values of R and S. Because this family provides a solution for any choices of a_0 and a_1, it contains all the solutions.

Because |1/φ|<1, we find that asymptotically a_n ~ Rφ^n as n grows. Therefore the ratio of terms tends to φ in the limit.