GATE | GATE CS 2010 | Question 5

What is the value of Lim_n->∞(1-1/n)²ⁿ ?
(A) 0
(B) e^-2
(C) e^-1/2
(D) 1

Answer: (B)

Explanation: The value of e (mathematical constant) can be written as following
$\lim_{n\rightarrow \infty }\left ( 1 + \frac{1}{n} \right )^{n}$

And the value of 1/e can be written as following.

$\frac{1}{e}=\lim_{n\rightarrow \infty }\left ( 1 - \frac{1}{n} \right )^{n}$

$\lim_{n\rightarrow \infty }\left ( 1 - \frac{1}{n} \right )^{2n} = \left (\lim_{n\rightarrow \infty }\left ( 1 - \frac{1}{n} \right )^{n} \right )^{2} = e^{-2}$

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