Evaluate the limit as n approaches infinity of (n/(n+1))^n.

Evaluate the limit as n approaches infinity of (n/(n+1))^n.

• A. 0
• B. e^{-1}
• C. 1
• D. e

Answer: (B)

Think of it as a standard “atomizes to e” scenario: a base that’s close to 1 raised to a large power. Write the expression as (1 - 1/(n+1))^n. Taking natural logs, let L be the limit and compute ln L = lim n ln(1 - 1/(n+1)). For small t, ln(1 - t) ~ -t, so with t = 1/(n+1) we get ln L = lim n * (-1/(n+1)) = -1. Therefore L = e^{-1}. Another way is to note (1 - 1/(n+1))^{n+1} -> e^{-1} and the overall exponent n/(n+1) -> 1, giving the same result.

Think of it as a standard “atomizes to e” scenario: a base that’s close to 1 raised to a large power. Write the expression as (1 - 1/(n+1))^n. Taking natural logs, let L be the limit and compute ln L = lim n ln(1 - 1/(n+1)). For small t, ln(1 - t) ~ -t, so with t = 1/(n+1) we get ln L = lim n * (-1/(n+1)) = -1. Therefore L = e^{-1}. Another way is to note (1 - 1/(n+1))^{n+1} -> e^{-1} and the overall exponent n/(n+1) -> 1, giving the same result.