lim(x→∞)[∫(0积到x）（t²乘以e的t²次方）dt]/[x乘以e的x²次方]=?

题目

lim(x→∞)[∫(0积到x）（t²乘以e的t²次方）dt]/[x乘以e的x²次方]=?

答案

∵lim(x→∞) ∫(0→x) [t²e^(t²)]dt→∞
lim(x→∞) [xe^(x²)]→∞
∴原式为∞/∞型未定式,用L'Hospital法则,分子分母同时对x求导
原式=lim(x→∞) { ∫(0→x) [t²e^(t²)]dt}'/[xe^(x²)]'
=lim(x→∞) [x²e^(x²)]/[e^(x²)+2x²e^(x²)]
=lim(x→∞) x²/(1+2x²)
=lim(x→∞) 1/[(1/x²)+2]
=1/2