求极限,x趋于1的,分子是1-根号下x,分母是1-立方根号x

问题描述:

求极限,x趋于1的,分子是1-根号下x,分母是1-立方根号x

方法一:
lim(x→1){[1-x^(1/2)]/[1-x^(1/3)]}
=lim(x→1){[1-x^(3/6)]/[1-x^(2/6)]}
=lim(x→1){[1+x^(1/6)+x^(2/6)]/[1+x^(1/6)]}
=[1+1^(1/6)+1^(2/6)]/[1+1^(1/6)]
=3/2
方法二:利用洛必塔法则.
lim(x→1){[1-x^(1/2)]/[1-x^(1/3)]}
=lim(x→1){[-(1/2)x^(-1/2)]/[-(1/3)x^(-2/3)]}
=[-(1/2)×1^(-1/2)]/[-(1/3)×1^(-2/3)]
=3/2.