I'm attempting to answer for the "Mathematica" listed above.
f[a_, x_] := a (ArcTan[x] + 2);(*x<0*)
g[b_, x_] := 2 Exp[b x] + 1; (*0<=x<=3*)
h[x_] := Log[x - 2] + x^2; (*x>3*)
solb = b /. Solve[ Limit[g[b, x], x -> 3] == h[3], b][[1]]
sola = a /. Solve[ Limit[ f[a, x], x -> 0] == g[b, 0], a][[1]]
Unfortunately the values I get namely a=Log[4]/3=0.46
and b=3/2
do not result in the three pieces having the same slope at the "joins". Enforcing the same slope at the two joins gave a=1.18
and b=0.59
. Therefore I don't think there is a solution. Maybe someone else will post a better solution.