limx→1 [x^(1/2)-1]/[x^(1/3)-1]
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limx→1[x^(1/2)-1]/[x^(1/3)-1]limx→1[x^(1/2)-1]/[x^(1/3)-1]limx→1[x^(1/2)-1]/[x^(1/3)-1]给出三种方法:(1)用洛必
limx→1 [x^(1/2)-1]/[x^(1/3)-1]
limx→1 [x^(1/2)-1]/[x^(1/3)-1]
limx→1 [x^(1/2)-1]/[x^(1/3)-1]
给出三种方法:
(1)用洛必达法则可得原式=lim(1/2√x)/(x^(-2/3)/3)=3/2
(2)设u=1+x
则原式=lim(u->0)[(1+u)^(1/2)-1]/[(1+u)^(1/3)-1]
由泰勒展式,原式=lim [(1/2)u+o(u)]/[(1/3)u+o(u)]=3/2
(3)原式=lim(u->0)[(1+u)^(1/2)-1]/[(1+u)^(1/3)-1]
分子分母有理化
分子分母同乘[(1+u)^(1/2)-1][(1+u)^(2/3)+(1+u)^(1/3)+1]
原式=lim [u * ((1+u)^(2/3)+(1+u)^(1/3)+1)] / [ u * ((1+u)^(1/2)+1)]
=3/2
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