Question

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

Answer 1

∵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