4. Which of the following functions grow faster than x² as x→∞? Which grow at the same rate as x²? Which grow slower?
e. x^3 - x^2

Verified step by step guidance

1. Identify the dominant term in the function \(x^3 - x^2\) as \(x \to \infty\). The dominant term is the one with the highest power of \(x\), which is \(x^3\) in this case.
2. Recall that the growth rate of a function as \(x \to \infty\) is determined by its highest power term. Here, \(x^3\) grows faster than \(x^2\) because the exponent 3 is greater than 2.
3. Compare the dominant term \(x^3\) with \(x^2\): since \(x^3\) grows faster than \(x^2\), the function \(x^3 - x^2\) grows faster than \(x^2\) as \(x \to \infty\).
4. To confirm, consider the limit \(\lim_{x \to \infty} \frac{x^3 - x^2}{x^2} = \lim_{x \to \infty} (x - 1) = \infty\), which shows the function grows faster than \(x^2\).
5. Therefore, \(x^3 - x^2\) grows faster than \(x^2\) as \(x\) approaches infinity.

Key Concepts

Asymptotic Growth Rates

Asymptotic growth rates describe how functions behave as the input variable approaches infinity. Comparing growth rates helps determine which functions increase faster, slower, or at the same rate by focusing on dominant terms and ignoring lower-order terms.

Dominant Term in Polynomials

In polynomial functions, the term with the highest exponent dominates the function's behavior for large values of x. For example, in x³ - x², the x³ term grows faster than x², so the overall function behaves like x³ as x approaches infinity.

Big-O and Big-Theta Notation

Big-O and Big-Theta notations classify functions by their growth rates. Big-O gives an upper bound, while Big-Theta indicates tight bounds, meaning two functions grow at the same rate. These notations help compare functions like x² and x³ in terms of their growth as x→∞.