求下列极限
1 lim(x→0)∫(0→x)arctantdt/x^2 2 lim(x→0)∫(0→sinx)sint^2dt/x^3 3 lim(x→0)∫(0→x)tln(1+t^2)dt/∫(0→x) (e^2-1)t^2dt
用罗比达法则、变上限定积分函数的求导法则, 还要用到x→0时的如下几个等价无穷小: arctanx~x; sin[sinx]~sinx~x ln(1+x^2)~x^2 e^x-1~x. 1】 lim(x→0)∫(0→x)arctantdt/x^2 =lim(x→0)[arctanx/(2x)]=1/2. 2】 lim(x→0)∫(0→sinx)sint^2dt/x^3 =lim(x→0)(cosx)[sin(sinx)]^2/(3x^2) =1/3. 3】 lim(x→0)∫(0→x)tln(1+t^2)dt/∫(0→x)(e^t-1)t^2dt =lim(x→0) x*ln(1+x^2)/(e^x-1)*x^2 =lim(x→0) x*x^2)/(x*x^2)=1.
答:详情>>