Question

lim（x→0）（∫（0，x）e^t²dt）²/∫（0，x）te^（2t³）dt

Answer 1

lim(x->0) [∫(0->x) e^(t^2) dt]^2/∫(0->x) te^(2t^3) dt
(0/0 分子分母分别求导）
=lim(x->0) 2.e^(t^2). ∫(0->x) e^(t^2) dt / [xe^(2x^3)]
=lim(x->0) 2∫(0->x) e^(t^2) dt / x (0/0 分子分母分别求导）
=lim(x->0) 2e^(x^2)
=2

Answer 2

x趋近0时 lim(x^2f(x)-f(2x^3))/x^3 =lim(x->0)f(x)/x-lim(x->0)f(2x^3)/x^3 =lim(x->0)[f(x)-f(0)]/x-lim(x->0)[f(2x^3)-f(0)]/x^3 =f'(0)-2 lim(x->0)[f(2x^3)-f(0)]/2x^3 =f'(0)-2f'(0) =-f'(0)