Evaluate lim x→0 (1 − e^{−x})/x.

Evaluate lim x→0 (1 − e^{−x})/x.

• A. 0
• B. ∞
• C. 1
• D. -1

Answer: (C)

As x approaches zero, the exponential behaves smoothly, and we can use a small-x expansion: e^{-x} ≈ 1 - x. This makes the numerator 1 - e^{-x} ≈ x, so the whole fraction approaches x/x = 1. A rigorous way is to apply L’Hôpital’s rule because the expression is 0/0 at x = 0: the derivative of the numerator 1 - e^{-x} is e^{-x}, and the derivative of the denominator is 1, giving e^{-x} → e^0 = 1. Therefore, the limit is 1. The other values don’t fit this behavior, since the ratio tends to a finite number (not 0, ∞, or -1) based on the exponential’s linear approximation near zero.

As x approaches zero, the exponential behaves smoothly, and we can use a small-x expansion: e^{-x} ≈ 1 - x. This makes the numerator 1 - e^{-x} ≈ x, so the whole fraction approaches x/x = 1. A rigorous way is to apply L’Hôpital’s rule because the expression is 0/0 at x = 0: the derivative of the numerator 1 - e^{-x} is e^{-x}, and the derivative of the denominator is 1, giving e^{-x} → e^0 = 1. Therefore, the limit is 1. The other values don’t fit this behavior, since the ratio tends to a finite number (not 0, ∞, or -1) based on the exponential’s linear approximation near zero.