First slide

Introduction to limits

Question

$lim_{n \to \infty} {(\frac{n^{2} - n + 1}{n^{2} - n - 1})}^{n (n - 1)}$ is equal to

Moderate

A

e
B

e²
C

e^-1
D

1

Solution

$\begin{matrix} lim_{n \to \infty} {(\frac{n^{2} - n + 1}{n^{2} - n - 1})}^{n (n - 1)} \\ = lim_{n \to \infty} {(\frac{n (n - 1) + 1}{n (n - 1) - 1})}^{n (n - 1)} \\ = lim_{n \to \infty} \frac{[n (n - 1)]^{n (n - 1)} {[1 + \frac{1}{n (n - 1)}]}^{n (n - 1)}}{[n (n - 1)]^{n (n - 1)} {[1 - \frac{1}{n (n - 1)}]}^{n (n - 1)}} \\ = lim_{n \to \infty} \frac{{(1 + \frac{1}{n (n - 1)})}^{n (n - 1)}}{{(1 - \frac{1}{n (n - 1)})}^{n (n - 1)}} = \frac{e}{e^{- 1}} = e^{2} [∵ lim_{n \to \infty} {(1 + \frac{1}{n})}^{n} = e] \end{matrix}$

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