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  4. lim limitsx→∞((n2-n+1/n2 - n - 1))n(n-1) is equal to

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Q. $\lim\limits_{x\to\infty}\left(\frac{n^{2}-n+1}{n^{2} - n - 1}\right)^{n\left(n-1\right)}$ is equal to

Limits and Derivatives

Solution:

$\lim\limits_{x\to\infty}\left(\frac{n^{2}-n+1}{n^{2} - n - 1}\right)^{n\left(n-1\right)} = \lim\limits_{n\to\infty}\left(\frac{n\left(n-1\right)+1}{n\left(n-1\right)-1}\right)^{n\left(n-1\right)} $
$=\lim\limits_{x\to\infty} \frac{\left(1+\frac{1}{n\left(n-1\right)}\right)^{n\left(n-1\right)}}{\left(1-\frac{1}{n\left(n-1\right)}\right)^{n\left(n-1\right)} } = \frac{e}{e^{-1}} = e^{2}$