4.1

eq = 1 - 1/(1 - 1/(1 - 1/(1 - 1/x)))

Expand[Simplify[eq]]

4.2

fx = (x^2)/(((Sqrt[1 + 2  x]) - (Sqrt[1 - 2  x]))*Sin[x])

Limit[fx, x -> 0]

Plot[fx, {x, -0.5, 0.5}, PlotRange ->  {0, 1}]

4.3

fx = f1 x + f0
gt = g1 t + g0

ans = 
      Simplify[(∫_0^1)(fx /. x -> x - t)*
              gt \[DifferentialD]t - fx]

kf0 = Coefficient[ans, f0]
kf1 = Coefficient[ans, f1]

kf10 = Coefficient[kf1, x, 0]
kf11 = Coefficient[kf1, x]

Solve[{kf0 == 0, kf10 == 0, kf11 == 0}, {g0, 
        g1}]

4.4

s = 200!

n = 0

While[Mod[s, 10] == 0, {s = s/10, n = n + 1}]

Print[n]

4.5

fx = ((2 x - 1) (2 x + 1))^6

fx1 = Expand[fx]

Coefficient[fx1, x^6]

4.6

fx = a x^2 + b x + c

fx1 = (fx /. x ->  x + 2) - fx

eq = {(fx /. x -> 0) == 1, 
        Coefficient[fx1, x, 0] == 0, 
        Coefficient[fx1, x] == 4}

Solve[eq, {a, b, c}]

4.7

eq = (p x + q) /. p -> (1 - 3 q)/3

Solve[{y == Coefficient[eq, q, 0], 
        0 == Coefficient[eq, q]}, {x, y}]

4.8

Mod[3^100, 7]

4.9

w = (Sqrt[3]) (1 + I) + (1 - I)

Expand[w^2]

Simplify[Abs[w^2]]

Simplify[Arg[w^2]]

Do[If[Mod[15 * n , 180] == 0, 
        If[Mod[15*n, 360] == 0, , Print[n]]] , {n, 1, 
        100} ]