lim(x→1) (x^n-1)/(x^m-1) m,n为正整数 在不用洛必达法则时怎么求
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n, m 是正整数
x^n -1 = (x-1)[ x^(n-1)+x^(n-2)+...+1]
x^m -1 = (x-1)[ x^(m-1)+x^(m-2)+...+1]
lim(x->1) (x^n -1)/(x^m -1)
=lim(x->1) (x-1)[ x^(n-1)+x^(n-2)+...+1] /{ (x-1)[ x^(m-1)+x^(m-2)+...+1] }
=lim(x->1) [ x^(n-1)+x^(n-2)+...+1] /[ x^(m-1)+x^(m-2)+...+1]
=n/m
x^n -1 = (x-1)[ x^(n-1)+x^(n-2)+...+1]
x^m -1 = (x-1)[ x^(m-1)+x^(m-2)+...+1]
lim(x->1) (x^n -1)/(x^m -1)
=lim(x->1) (x-1)[ x^(n-1)+x^(n-2)+...+1] /{ (x-1)[ x^(m-1)+x^(m-2)+...+1] }
=lim(x->1) [ x^(n-1)+x^(n-2)+...+1] /[ x^(m-1)+x^(m-2)+...+1]
=n/m
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