Question

求极限lim(x→0)∫(x,0)(xcost^2dx)/x

Answer 1

说明：此题是打错了！我想应该是：求极限lim(x→0)∫(x,0)(tcost^2dt)/x.
若是这样，解法如下。
解法一：原式=lim(x→0)[xcos(x²)] (0/0型，应用罗比达法则)
=0*1
=0；
解法二：原式=lim(x→0){[1/2∫(x,0)cos(t²)d(t²)]/x}
=lim(x→0){[sin(x²)/2]/x}
=lim(x→0){[(x/2)*[sin(x²)/(x²)]}
=lim(x→0){[(x/2)*lim(x→0)[sin(x²)/(x²)]
=0*1 (应用重要极限lim(x→0)（sinx/x)=1)
=0.