How to find limit of a recursive sequence?

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In your example all f(n) are equal to 1, so the limit is 1 :)

In the case of linear recursive definitions you can use rsolve from sympy or solve_rec from Maxima, and then use limit function. In the general nonlinear case you can experiment numerically: http://ask.sagemath.org/question/1247....

Using Maxima you can do:

sage: maxima('f[n]:=1/5*(f[n-1]^2+f[n-1]+3)')
f[n]:=(3+f[n-1]+f[n-1]^2)/5
sage: maxima('f[0]:1')                       
1
sage: maxima('f[100]')                       
1