3. Which of the following functions grow faster than x² as x→∞? Which grow at the same rate as x²? Which grow slower?
g. x^3 e^(-x)

Verified step by step guidance

1. Identify the function given: \(g(x) = x^{3} e^{-x}\).
2. Recall that as \(x \to \infty\), the exponential term \(e^{-x}\) approaches 0 very rapidly, while the polynomial term \(x^{3}\) grows without bound but at a much slower rate compared to exponentials.
3. Analyze the behavior of \(g(x)\) by considering the dominant terms: since \(e^{-x}\) decays faster than any polynomial grows, the product \(x^{3} e^{-x}\) tends to 0 as \(x \to \infty\).
4. Compare this behavior to \(x^{2}\): since \(g(x)\) tends to 0 and \(x^{2}\) tends to infinity, \(g(x)\) grows slower than \(x^{2}\) as \(x \to \infty\).
5. Conclude that \(g(x)\) grows slower than \(x^{2}\) as \(x\) approaches infinity.

Key Concepts

Asymptotic Growth Rates

Asymptotic growth rates describe how functions behave as the input grows very large. Comparing growth rates helps determine which functions increase faster, slower, or at the same rate as a reference function, such as x², by analyzing dominant terms and limits at infinity.

Exponential Decay vs Polynomial Growth

Exponential decay functions like e^(-x) decrease rapidly to zero as x approaches infinity, often overpowering polynomial growth terms. When combined, the exponential decay can cause the entire function to approach zero, affecting the overall growth rate compared to pure polynomials.

Limit Comparison for Growth Rates

To compare growth rates, we use limits of the ratio of two functions as x approaches infinity. If the limit is zero, the numerator grows slower; if infinite, it grows faster; if finite and nonzero, they grow at the same rate. This method rigorously classifies growth behavior.