Evaluate lim x→∞ x e^{−x}.

Evaluate lim x→∞ x e^{−x}.

• A. 0
• B. ∞
• C. 1
• D. -1

Answer: (A)

Exponential decay dominates linear growth. As x grows, e^{−x} shrinks to zero much faster than x grows, so their product x e^{−x} tends to zero. A clear way to see this is rewrite as x / e^{x}. The numerator grows only linearly, while the denominator grows exponentially. Formally, applying L’Hôpital’s rule to lim_{x→∞} x / e^{x} gives lim_{x→∞} 1 / e^{x} = 0. Therefore, the limit is 0.

Exponential decay dominates linear growth. As x grows, e^{−x} shrinks to zero much faster than x grows, so their product x e^{−x} tends to zero.

A clear way to see this is rewrite as x / e^{x}. The numerator grows only linearly, while the denominator grows exponentially. Formally, applying L’Hôpital’s rule to lim_{x→∞} x / e^{x} gives lim_{x→∞} 1 / e^{x} = 0. Therefore, the limit is 0.