What is the value of lim_{x-inf} (1 - 1/(2x))^x ?

A. 0
B. e^(-1/2)
C. e^(1/2)
D. 1

Question

What is the value of lim_{x->inf} (1 - 1/(2x))^x ?

A. 0
B. e^(-1/2)
C. e^(1/2)
D. 1

Accepted Answer

e^(-1/2) This is a standard 1^infty indeterminate form. The trick is to write the expression as e raised to the power x*ln(1 - 1/(2x)), then use the approximation ln(1+u) ≈ u for small u near 0. As x -> inf, 1/(2x) -> 0 so u = -1/(2x) is small, and x * (-1/(2x)) = -1/2. The limit is e^(-1/2).

What is the value of lim_{x->inf} (1 - 1/(2x))^x ? A. 0 B. e^(-1/2) C. e^(1/2) D. 1