_ x 1 x^m-1 x^n-1 is equal to _________________ - Vidyadip

Question

$\lim _{x \rightarrow 1} \frac{x^m-1}{x^n-1}$ is equal to _________________

✓

Answer

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

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