Question

高数计算,求详解lim(x->∞){ [e^(-x^2)]*[∫(0->x)(t^2)*e^(t^2) dt]}/x,

Answer 1

原式分子中的[e^(-x^2)]放到分母中得

原式=lim(x->∞)[∫(0->x)(t^2)*e^(t^2) dt]/[xe^(x^2)],用洛必达法则分子分母分别求导得

=lim(x->∞)[x^2)*e^(x^2)]/[e^(x^2)+(2x^2)*e^(x^2)],化简消去e^(x^2)得

=lim(x->∞)[x^2]/[1+(2x^2)]=1/2.