Evaluate the limit as x approaches infinity of e^{-x} x^k for a fixed k.

Evaluate the limit as x approaches infinity of e^{-x} x^k for a fixed k.

• A. 0
• B. 1
• C. Infinity
• D. k

Answer: (A)

Exponential decay dominates any polynomial growth. Rewriting the expression as x^k / e^{x}, the denominator grows far faster than the numerator as x goes to infinity, so the ratio tends to zero. If you apply L’Hôpital’s rule k times, you get k! / e^{x} in the end, which clearly tends to zero, confirming the limit. This holds for any fixed real k (even negative), since the exponential term always outpaces a polynomial of fixed degree.

Exponential decay dominates any polynomial growth. Rewriting the expression as x^k / e^{x}, the denominator grows far faster than the numerator as x goes to infinity, so the ratio tends to zero. If you apply L’Hôpital’s rule k times, you get k! / e^{x} in the end, which clearly tends to zero, confirming the limit. This holds for any fixed real k (even negative), since the exponential term always outpaces a polynomial of fixed degree.