Question

问个数学极限方面的题：求极限lim x^n-1/x-1 (n 为正整数)x->1请详细解释作题的基础知识,特别是 x的n

Answer 1

x^n-1

=(x^n+x^(n-1)+x^(n-2)+x^(n-3)+...+x)-(x^(n-1)+x^(n-2)+x^(n-3)+...+x+1)

=x(x^(n-1)+x^(n-2)+x^(n-3)+...+x+1)-(x^(n-1)+x^(n-2)+x^(n-3)+...+x+1)

=(x-1)(x^(n-1)+x^(n-2)+x^(n-3)+...+x+1)

两边同时除以(x-1)得

(x^n-1)/(x-1)=x^(n-1)+x^(n-2)+x^(n-3)+...+x+1

两边取极限得

lim (x^n-1)/(x-1)=n (x->1)

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当然,如果用高等数学的洛必达法则求解会更简捷

lim (x^n-1)/(x-1) (x->1)

=lim (x^n-1)'/(x-1)'

=lim n*(x^(n-1))/1

=n*lim x^(n-1)

=n