Evaluate _ x 1 x^m-1 x^n-1 . - Vidyadip

Question

Evaluate $\lim _{x \rightarrow 1} \frac{x^m-1}{x^n-1}.$

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Answer

$\begin{aligned} \lim _{x \rightarrow 1} \frac{x^m-1}{x^n-1} & \\ & =\lim _{x \rightarrow 1} \frac{x^m-1}{x-1} \cdot \frac{x-1}{x^n-1} \\ & =\lim _{x \rightarrow 1} \frac{x^m-1}{x-1} \cdot \frac{1}{\frac{x^n-1}{x-1}} \\ & =\lim _{x \rightarrow 1} \frac{x^m-1}{x-1} \times \lim _{x \rightarrow 1} \frac{1}{\frac{x^n-1}{x-1}}=\frac{m(1)^{m-1}}{m(1)^{n-1}} \\ & =\frac{m}{n} \quad\left[\because \lim _{x \rightarrow a} \frac{x^m-a^m}{x-a}=m a^{m-1}\right]\end{aligned}$

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