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

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

• A. e
• B. 1
• C. ∞
• D. 0

Answer: (A)

Taking natural log helps reveal why this limit equals e. Let L be the limit of (1 + 1/x)^x as x → ∞. Then ln L = lim x ln(1 + 1/x). For small 1/x, the log expansion gives ln(1 + t) ≈ t, so x ln(1 + 1/x) ≈ x*(1/x) = 1, and the remaining terms vanish as x grows. Thus ln L = 1, so L = e. This is the familiar limit that defines e. The expression doesn’t approach 1, ∞, or 0; it converges to the finite value e.

Taking natural log helps reveal why this limit equals e. Let L be the limit of (1 + 1/x)^x as x → ∞. Then ln L = lim x ln(1 + 1/x). For small 1/x, the log expansion gives ln(1 + t) ≈ t, so x ln(1 + 1/x) ≈ x*(1/x) = 1, and the remaining terms vanish as x grows. Thus ln L = 1, so L = e. This is the familiar limit that defines e. The expression doesn’t approach 1, ∞, or 0; it converges to the finite value e.