a) x=√(3cos(x)+10).
If we start with x₀=0, x₁=√13 radians
x₁=√(3cos(√13)+10)=2.705020485,
x₂=2.698403383,
x₃=2.69996928,
x₄=2.699596815,
x₅=2.699685303,
x₆=2.699664274,
x₇=2.699669271,
x₈=2.699668084.
The error is now <0.00001. x=2.69967
b) x=(3cos(x)+10)/x.
x₀=2.5,
x₁=3.038627661,
x₂=2.30890406,
x₃=3.456771073,...
The values are diverging.
c) arccos((x²-10)/3).
Since -1≤(x²-10)/3≤1, -3+10≤x²≤3+10, √7≤x≤√13,
x₀=√7,
x₁=π,
x₂=1.614275224 leading to a mathematical error for the next iteration.
The same result applies if x₀=√13, after a few divergent iterations.