Mathematics

Mathematics

Problem

Let h(x)=5x3+1exh(x)=\dfrac{5x^{3}+1}{e^{x}}. Which of the following best explains the long-run behavior of hh? a) hh has no horizontal asymptote because the numerator is a higher-degree polynomial than the denominator. b) hh has a horizontal asymptote at y=5y=5 because the leading coefficient of the numerator determines the end behavior. c) hh has a horizontal asymptote at y=0y=0 because the exponential in the denominator eventually grows faster than any polynomial numerator. d) hh has a horizontal asymptote at y=0y=0 because as x→−∞x\to-\infty the denominator exe^{x} approaches 00."
Intuitive explanation

Exponential functions grow much faster than polynomials, so even though the numerator 5x3+15x^3 + 1 gets arbitrarily large, the denominator exe^x eventually dominates it completely. The key insight is to recognize that any polynomial divided by an exponential function will approach zero as x→∞x \to \infty, because exponential growth always wins in the long run.

Intuitive explanation

Exponential functions grow much faster than polynomials, so even though the numerator 5x3+15x^3 + 1 gets arbitrarily large, the denominator exe^x eventually dominates it completely. The key insight is to recognize that any polynomial divided by an exponential function will approach zero as x→∞x \to \infty, because exponential growth always wins in the long run.

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