4. Which of the following functions grow faster than x² as x→∞? Which grow at the same rate as x²? Which grow slower?
c. x²e^(-x)

Verified step by step guidance

1. Identify the given function: \(f(x) = x^{2} e^{-x}\).
2. Recall that \(e^{-x}\) can be rewritten as \(\frac{1}{e^{x}}\), which decreases very rapidly as \(x \to \infty\).
3. Compare the growth rates: \(x^{2}\) grows polynomially, while \(e^{-x}\) decays exponentially, so their product \(x^{2} e^{-x}\) tends to zero as \(x \to \infty\).
4. Since \(x^{2} e^{-x}\) approaches zero, it grows slower than \(x^{2}\) as \(x \to \infty\).
5. Conclude that \(f(x) = x^{2} e^{-x}\) grows slower than \(x^{2}\) when \(x\) becomes very large.

Key Concepts

Growth Rates of Functions

Growth rates describe how functions behave as the input becomes very large. Comparing growth rates helps determine which functions increase faster, slower, or at the same pace as a reference function, such as x², by analyzing their dominant terms as x approaches infinity.

Exponential Decay vs Polynomial Growth

Exponential decay functions like e^(-x) decrease rapidly to zero as x increases, often overpowering polynomial growth terms. When combined, such as in x²e^(-x), the exponential decay dominates, causing the overall function to approach zero faster than any polynomial grows.

Limits and Asymptotic Behavior

Evaluating limits as x approaches infinity reveals the long-term behavior of functions. By calculating the limit of the ratio of two functions, we can determine if one grows faster, slower, or at the same rate as the other, which is essential for classifying growth rates.