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Q. $\displaystyle \lim_{x \to 1}$ $\frac{x^{m}-1}{x^{n}-1}$ is

Limits and Derivatives

Solution:

We have, $\displaystyle \lim_{x \to 1}$ $\frac{x^{m}-1}{x^{n}-1}$
$=\displaystyle \lim_{x \to 1}$ $\left\{\frac{x^{m}-1^{m}}{x-1}\times \frac{x-1}{x^{n}-1^{n}}\right\}$
$=\displaystyle \lim_{x \to 1} \left\{\frac{x^m-1^m}{x-1}\right\}$$\times \displaystyle \lim_{x \to 1}$ $\left\{\frac{x-1}{x^{n}-1^{n}}\right\}$
$=\frac{m\left(1\right)^{m-1}}{n\left(1\right)^{n-1}}=\frac{m}{n}$