题目

lim(x->1) x^(1/（1-x）) lim(x->0) x^(tanx)的值

提问时间：2021-03-12

答案

1.lim(x->1)[x^(1/(1-x))]=?
∵lim(x->1)[lnx/(1-x)]=lim(x->1)[(1/x)/(-1)] (0/0型极限,应用罗比达法则)
=lim(x->1)[(-1)/x]
=-1
∴lim(x->1)[x^(1/(1-x))]=lim(x->1){e^[lnx/(1-x)]}
=e^{lim(x->1)[lnx/(1-x)]}
=e^(-1)
=1/e.
2.lim(x->0)[x^(tanx)]=?
∵lim(x->0)(tanx*lnx)=lim(x->0)[(sinx/x)(1/cosx)(lnx/(1/x))]
=[lim(x->0)(sinx/x)]*[lim(x->0)(1/cosx)]*[lim(x->0)(lnx/(1/x))]
=1*1*[lim(x->0)(lnx/(1/x))] (应用重要极限)
=lim(x->0)(lnx/(1/x))
=lim(x->0)[(1/x/(-1/x²))] (∞/∞型极限,应用罗比达法则)
=lim(x->0)(-x)
=0
∴lim(x->0)[x^(tanx)]=lim(x->0)[e^(tanx*lnx)]
=e^[lim(x->0)(tanx*lnx)]
=e^(0)
=1.