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)

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Identify the given function: \(f(x) = x^{2} e^{-x}\).

Recall that \(e^{-x}\) can be rewritten as \(\frac{1}{e^{x}}\), which decreases very rapidly as \(x \to \infty\).

Compare the growth rates: \(x^{2}\) grows polynomially, while \(e^{-x}\) decays exponentially, so their product \(x^{2} e^{-x}\) tends to zero as \(x \to \infty\).

Since \(x^{2} e^{-x}\) approaches zero, it grows slower than \(x^{2}\) as \(x \to \infty\).

Conclude that \(f(x) = x^{2} e^{-x}\) grows slower than \(x^{2}\) when \(x\) becomes very large.

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