lim(x趋近零)[∫(1+t^2) e^(t^2-x^2)d(x)]/x^2 {定积分上限是x^2,下限为0}
问题描述：
lim(x趋近零)[∫(1+t^2) e^(t^2-x^2)d(x)]/x^2 {定积分上限是x^2,下限为0}
答
先整理分子,将带x的拿到积分外
∫(1+t^2) e^(t^2-x^2)d(x)=e^(-x^2)∫(1+t^2) e^(t^2)d(x)
然后用洛必达法则
原式=lim(x趋近零)e^(-x^2)∫(1+t^2) e^(t^2)d(x)/x^2
=lim(x趋近零)e^(-x^2)*lim∫(1+t^2) e^(t^2)d(x)/x^2
=1*lim(1+x^4) e^(x^4)/2x
=无穷大