Compute the limit as x approaches infinity of (1 + 1/x)^x.

Compute the limit as x approaches infinity of (1 + 1/x)^x.

• A. 2.5
• B. 3
• C. 0
• D. e

Answer: (D)

This limit ties directly to the number e, the base of natural logarithms. Take natural logs: if L = lim_{x→∞} (1 + 1/x)^x, then ln L = lim_{x→∞} x ln(1 + 1/x). For small 1/x, ln(1 + 1/x) ≈ 1/x, so x ln(1 + 1/x) ≈ x*(1/x) = 1. Therefore ln L = 1, which gives L = e. Since e ≈ 2.718, this value is between 2.5 and 3, not 0 or any other option listed.

This limit ties directly to the number e, the base of natural logarithms. Take natural logs: if L = lim_{x→∞} (1 + 1/x)^x, then ln L = lim_{x→∞} x ln(1 + 1/x). For small 1/x, ln(1 + 1/x) ≈ 1/x, so x ln(1 + 1/x) ≈ x*(1/x) = 1. Therefore ln L = 1, which gives L = e. Since e ≈ 2.718, this value is between 2.5 and 3, not 0 or any other option listed.