Evaluate lim x→0 (1 − e^{−x})/x.

Evaluate lim x→0 (1 − e^{−x})/x.

• A. 0
• B. -1
• C. 1
• D. 2

Answer: (C)

Near zero, the exponential behaves linearly: e^{−x} ≈ 1 − x. So 1 − e^{−x} ≈ x, and dividing by x gives a ratio that tends to 1. Equivalently, using the standard limit lim_{t→0} (e^{t} − 1)/t = 1 and setting t = −x, we get (1 − e^{−x})/x → 1 as x → 0.

Near zero, the exponential behaves linearly: e^{−x} ≈ 1 − x. So 1 − e^{−x} ≈ x, and dividing by x gives a ratio that tends to 1.

Equivalently, using the standard limit lim_{t→0} (e^{t} − 1)/t = 1 and setting t = −x, we get (1 − e^{−x})/x → 1 as x → 0.