Compute lim_{x->∞} x^2 e^{-x}.

Compute lim_{x->∞} x^2 e^{-x}.

• A. ∞
• B. 1
• C. 0
• D. -∞

Answer: (C)

Exponential growth in the denominator dominates any polynomial growth in the numerator, so the ratio goes to zero as x becomes large. Write the expression as x^2/e^x. Applying L’Hôpital’s rule: the first derivative gives 2x/e^x, and applying it again yields 2/e^x. Since e^x grows without bound, 2/e^x tends to 0 as x→∞. Therefore, the limit is 0. The expression is positive for large x, so it cannot go to ∞ or −∞, and it cannot approach a nonzero finite value like 1.

Exponential growth in the denominator dominates any polynomial growth in the numerator, so the ratio goes to zero as x becomes large. Write the expression as x^2/e^x. Applying L’Hôpital’s rule: the first derivative gives 2x/e^x, and applying it again yields 2/e^x. Since e^x grows without bound, 2/e^x tends to 0 as x→∞. Therefore, the limit is 0. The expression is positive for large x, so it cannot go to ∞ or −∞, and it cannot approach a nonzero finite value like 1.