4. Which of the following functions grow faster than x² as x→∞? Which grow at the same rate as x²? Which grow slower?
a. x² + √x

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Identify the dominant term in the function as \( x \to \infty \). For the function \( f(x) = x^{2} + \sqrt{x} \), the terms are \( x^{2} \) and \( \sqrt{x} = x^{1/2} \).

Compare the growth rates of each term to \( x^{2} \). Since \( x^{2} \) grows faster than \( x^{1/2} \), the \( x^{2} \) term dominates the behavior of the function for large \( x \).

Determine if the function grows faster, slower, or at the same rate as \( x^{2} \) by considering the limit \( \lim_{x \to \infty} \frac{f(x)}{x^{2}} = \lim_{x \to \infty} \frac{x^{2} + \sqrt{x}}{x^{2}} \).

Simplify the limit expression to \( \lim_{x \to \infty} \left(1 + \frac{\sqrt{x}}{x^{2}}\right) = \lim_{x \to \infty} \left(1 + x^{-3/2}\right) \).

Evaluate the limit: since \( x^{-3/2} \to 0 \) as \( x \to \infty \), the limit equals 1, indicating that \( f(x) \) grows at the same rate as \( x^{2} \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 pace as a reference function, such as x², by focusing on dominant terms and ignoring lower-order terms.

Dominant Term in Polynomial Functions

In polynomials, the term with the highest exponent dominates the function's behavior for large x. For example, in x² + √x, the x² term grows faster than √x as x→∞, so the overall growth rate is determined by x², making it grow at the same rate as x².

Big-O and Big-Theta Notation

Big-O and Big-Theta notations formalize growth comparisons by classifying functions based on upper bounds (Big-O) or tight bounds (Big-Theta). Saying a function grows at the same rate as x² means it is Θ(x²), while growing faster or slower corresponds to different asymptotic classes.

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