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

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Identify the dominant term in the function as \( x \to \infty \). For the function \( f(x) = x^2 + 4x \), the dominant term is \( x^2 \) because it grows faster than \( 4x \) when \( x \) becomes very large.

Compare the growth rate of \( f(x) \) to \( x^2 \) by considering the ratio \( \frac{f(x)}{x^2} = \frac{x^2 + 4x}{x^2} \).

Simplify the ratio: \( \frac{x^2 + 4x}{x^2} = 1 + \frac{4}{x} \).

Analyze the limit of the ratio as \( x \to \infty \): \( \lim_{x \to \infty} \left(1 + \frac{4}{x}\right) = 1 \). This means \( f(x) \) grows at the same rate as \( x^2 \).

Conclude that \( x^2 + 4x \) grows at the same rate as \( x^2 \) because the lower order term \( 4x \) becomes insignificant compared to \( x^2 \) for large \( x \).

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

Dominant Term in Polynomials

In polynomial functions, the term with the highest power of x dominates the function's behavior as x approaches infinity. For example, in x² + 4x, the x² term grows faster than 4x, so the overall growth rate is determined by x².

Big-O and Big-Theta Notation

Big-O and Big-Theta notations classify functions by their growth rates. Big-Theta (Θ) indicates functions grow at the same rate, while Big-O indicates an upper bound. These notations help formalize comparisons like whether a function grows faster, slower, or at the same rate as x².

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