Question

求下列极限答题步骤

Answer 1

（1）原式=lim(x->0)[(∫cos(t^2)dt)'/(x^2)'] (0/0型极限,应用罗比达法则)

=lim(x->0)[cos(x^4)]

=1；

（2）原式=lim(x->0)[((∫e^(t^2)dt)^2)'/(∫te^(2t^2)dt)'] (0/0型极限,应用罗比达法则)

=lim(x->0)[(2e^(x^2)∫e^(t^2)dt)/(xe^(2t^2))]

=lim(x->0)[(2/e^(x^2))((∫e^(t^2)dt)/x)]

={lim(x->0)[2/e^(x^2)]}*{lim(x->0)[(∫e^(t^2)dt)/x]}

=2*lim(x->0)[(∫e^(t^2)dt)/x]

=2*lim(x->0)[(∫e^(t^2)dt)'/(x)'] (0/0型极限,应用罗比达法则)

=2*lim(x->0)[e^(x^2)]

=2.