Question

Mọi người giúp em với.

Tính:

a) \(\lim\limits_{x\rightarrow1}\left(\frac{x^m-1}{x^n-1}\right)\)

b) \(\lim\limits_{x\rightarrow1}\frac{\left(1-\sqrt{x}\right)\left(1-\sqrt[3]{x}\right)\left(1-\sqrt[4]{x}\right)\left(1-\sqrt[5]{x}\right)}{\left(1-x\right)^4}\)

Answer 1

Nếu m,n là số tự nhiên :v

\(x^n-1=\left(x-1\right)\left(x^{n-1}+x^{n-2}+...+x+1\right)\)\(x^m-1=\left(x-1\right)\left(x^{m-1}+x^{m-2}+...+x+1\right)\)

\(\lim\limits_{x\rightarrow1}\left(\frac{x^m-1}{x^n-1}\right)=\lim\limits_{x\rightarrow1}\frac{\left(x-1\right)\left(x^{m-1}+...+x+1\right)}{\left(x-1\right)\left(x^{n-1}+...+x+1\right)}\)

\(=\frac{x^{m-1}+...+x+1}{x^{n-1}+....+x+1}=\frac{m}{n}\)