Evaluate the limit as x approaches infinity of (1 - 2/x)^x.

Evaluate the limit as x approaches infinity of (1 - 2/x)^x.

• A. e^{-1}
• B. e^{2}
• C. e^{0}
• D. e^{-2}

Answer: (D)

Think about how expressions of the form (1 + c/x)^x behave as x grows large. They tend to e^c because ln((1 + c/x)^x) = x ln(1 + c/x), and with u = c/x, ln(1 + u) ~ u for small u, so x ln(1 + c/x) ~ x*(c/x) = c. Exponentiating gives the limit e^c. Here the inside is 1 - 2/x, which is 1 + (-2)/x, so c = -2. Therefore the limit is e^{-2}. A quick justification uses ln(1+u) ~ u for small u: with u = -2/x, ln(1 - 2/x) ~ -2/x, so x ln(1 - 2/x) ~ -2, and exponentiating yields e^{-2}.

Think about how expressions of the form (1 + c/x)^x behave as x grows large. They tend to e^c because ln((1 + c/x)^x) = x ln(1 + c/x), and with u = c/x, ln(1 + u) ~ u for small u, so x ln(1 + c/x) ~ x*(c/x) = c. Exponentiating gives the limit e^c. Here the inside is 1 - 2/x, which is 1 + (-2)/x, so c = -2. Therefore the limit is e^{-2}. A quick justification uses ln(1+u) ~ u for small u: with u = -2/x, ln(1 - 2/x) ~ -2/x, so x ln(1 - 2/x) ~ -2, and exponentiating yields e^{-2}.