Consider the plane Hamiltonian system \[ dx/dt=- \partial H(x,y)/\partial y,\quad dy/dt=\partial H(x,y)/\partial x \] where \(H(x,y)\) is a real polynomial in \(x,y\). Many authors studied problems like isochronicity, monotonicity or bifurcation of critical period of a nondegerate center of the system (a center is said to be nondegerate if the linearized vector field at the point has at least one non-zero eigenvalue). In this paper, as a continuation of the authors' paper [J. Differ. Equations 180, No. 2, 334--373 (2002; Zbl 1014.34020)], the infinity behavior of the period function denoted by \(T(h)\), of the center is studied. Here, \(T(h)\) is a function of the period \(h\) of each periodic orbit inside the largest punctured neighborhood surrounding the center and when \(h\) changes on open interval \((0,a)\) such that the \(T\)-periodic orbit covers entirely the neighborhood annulus. The authors prove that in case \(n= 4\) the value \(T(h)\) tends to infinity as \(h\) tends to \(a\). In addition, an analytic expression of \(T(h)\) is obtained for \[ H(x,y)=A(x) + B(x)y + C(x)y*y +D(x)y*y*y \] to find the isochronicity conditions in this family.