limx→1 (3√x-1)/(√x-1)求极限
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解法一：原式=lim(x->1){[(x^(1/3)-1)(x^(2/3)+x^(1/3)+1)(√x+1)]/[(√x-1)(√x+1)(x^(2/3)+x^(1/3)+1)]}
=lim(x->1){[(x-1)(√x+1)]/[(x-1)(x^(2/3)+x^(1/3)+1)]}
=lim(x->1)[(√x+1)/(x^(2/3)+x^(1/3)+1)]
=(1+1)/(1+1+1)
=2/3；
解法二：原式=lim(x->1){[(1/3)x^(-2/3)]/[(1/2)x^(-1/2)]} (0/0型极限,应用罗比达法则)
=(1/3)/(1/2)
=2/3.
=lim(x->1){[(x-1)(√x+1)]/[(x-1)(x^(2/3)+x^(1/3)+1)]}
=lim(x->1)[(√x+1)/(x^(2/3)+x^(1/3)+1)]
=(1+1)/(1+1+1)
=2/3；
解法二：原式=lim(x->1){[(1/3)x^(-2/3)]/[(1/2)x^(-1/2)]} (0/0型极限,应用罗比达法则)
=(1/3)/(1/2)
=2/3.
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