GATE | GATE CS 2010 | Question 5

What is the value of Limn->∞(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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