Question

${\displaystyle \underset{x\to 1}{lim}\frac{x^m-1}{x^n-1}}$ is ______.

विकल्प

• 1

• ${\displaystyle \frac{m}{n}}$

• ${\displaystyle -\frac{m}{n}}$

• ${\displaystyle \frac{m^2}{n^2}}$

Answer 1

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

Explanation:

Given, ${\displaystyle \underset{x\to 1}{lim}\frac{x^m-1}{x^n-1}}$

= ${\displaystyle \underset{x\to 1}{lim}\frac{\frac{x^m-{\left(1\right)}^m}{x-1}}{\frac{x^n-{\left(1\right)}^n}{x-1}}}$

= ${\displaystyle \frac{m{\left(1\right)}^{m-1}}{n{\left(1\right)}^{n-1}}}$

= ${\displaystyle \frac{m}{n}}$  .....${\displaystyle [\because \underset{x\to a}{lim}\frac{x^n-a^n}{x-a}=n{a}^{n-1}]}$