Question

$\boxed{\sf \lim_{x \to 1} \dfrac{x^n-1}{x-1} }$


* Need a step-by-step answer!
* Thank you!
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Answer 1

$\begin{gathered}\\ \sf\Rrightarrow \lim_{x\to 1}\dfrac{x^n-1}{x-1}\end{gathered}$

$\begin{gathered}\\ \sf\Rrightarrow \lim_{x\to 1}\dfrac{(x-1)(x^{n-1}+x^{n-2}+\dots x^1+1)}{(x-1)}\end{gathered}$

• Cancel x-1

$\begin{gathered}\\ \sf\Rrightarrow \lim_{x\to 1}nx^{n-1}\end{gathered}$

• Apply limits

$\begin{gathered}\\ \sf\Rrightarrow n(1)^{n-1}\end{gathered}$

$\begin{gathered}\\ \sf\Rrightarrow n(1)\end{gathered}$

$\begin{gathered}\\ \sf\Rrightarrow n\end{gathered}$

$\begin{gathered} \sf\Rrightarrow \red n\end{gathered}$

${ \underline {\green{\rule51pt{8pt}}}}{ \underline {\red{\rule51pt{8pt}}}}{ \underline {\color{pink}{\rule51pt{8pt}}}}$

Rule Used :

• L' Hospital