Question

$\underset{x\to 1}{\lim }\frac{x^m-1}{x^n-1}$ ​is equal to ______________________.

Answer 1

$$\begin{gathered} \begin{array}{c}\lim \\ x\to 1\end{array}\frac{x^m-1}{x^n-1} \\ \mathrm{i}.\mathrm{e}.\begin{array}{c}\lim \\ x\to 1\end{array}\frac{x^m-1}{x-1}\times \frac{x-1}{x^n-1} \\ \mathrm{i}.\mathrm{e}.\left(\begin{array}{c}\lim \\ x\to 1\end{array}\frac{x^m-1}{x-1}\right)\left(\begin{array}{c}\lim \\ x\to 1\end{array}\frac{x-1}{x^n-1}\right) \\ \mathrm{i}.\mathrm{e}.m\times \frac{1}{n}\left(\because \begin{array}{c}\lim \\ x\to a\end{array}\frac{x^n-a}{x-a}=n{a}^{n-1}\right) \\ \mathrm{i}.\mathrm{e}.\frac{m}{n} \\ \mathrm{i}.\mathrm{e}.\begin{array}{c}\lim \\ x\to 1\end{array}\frac{x^m-1}{x^n-1}=\frac{m}{n} \end{gathered}$$