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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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.

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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.

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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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