When I called Wolfram|Alpha it gave the answer 1/4
. You can check that it is correct with an estimate:
f[x_, y_] = (x^2*(1 + Sin[y]) + y^2)/(4 x^2 + 4 y^2);
Plot3D[f[x, y],
{x, -1, 1}, {y, -1, 1}]
Reduce[RealAbs[f[x, y] - 1/4] <= RealAbs[x]/8, Reals]