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

Question

lim limitsx→∞((n2-n+1/n2 - n - 1))n(n-1) is equal to (A) e (B) e2 (C) e-1 (D) 1

Tardigrade Admin · Accepted Answer

lim limitsx→∞((n2-n+1/n2 - n - 1))n(n-1) = lim limitsn→∞((n(n-1)+1/n(n-1)-1))n(n-1) = lim limitsx→∞ ((1+(1/n(n-1)))n(n-1)/(1-(1)n(n-1))n(n-1) ) = (e/e-1) = e2

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

$e$

$e^{2}$

$e^{- 1}$

$1$

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

Q. x→∞lim​(n2−n−1n2−n+1​)n(n−1) is equal to

e

e2

e−1

1

x→∞lim​(n2−n−1n2−n+1​)n(n−1)=n→∞lim​(n(n−1)−1n(n−1)+1​)n(n−1) =x→∞lim​(1−n(n−1)1​)n(n−1)(1+n(n−1)1​)n(n−1)​=e−1e​=e2

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

$e$

$e^{2}$

$e^{- 1}$

$1$

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