f[t_]=t+2*Sin[t]+1;
Init=0;
For[{xg=Init,Print[n, "  ", xg,"  ",f[xg]],n=1},
Abs[f[xg]]>10^(-14) && (n<100) ,
n++,
{xg=N[xg-(f[xg])/f'[xg]],Print [n, "  ", SetPrecision[xg,20],"  ",SetPrecision[f[xg],20]],}]
Plot[f[x],{x,xg-.001,xg+.001}]