110. Does f grow faster, slower, or at the same rate as g as x→∞? Give reasons for your answers.
c. f(x) = 10x^3 + 2x^2, g(x) = e^x

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Identify the given functions: \(f(x) = 10x^{3} + 2x^{2}\) and \(g(x) = e^{x}\).

Recall that as \(x \to \infty\), polynomial functions like \(x^{3}\) grow at a rate proportional to a power of \(x\), while exponential functions like \(e^{x}\) grow much faster than any polynomial.

To compare growth rates, consider the limit of the ratio \(\frac{f(x)}{g(x)} = \frac{10x^{3} + 2x^{2}}{e^{x}}\) as \(x \to \infty\).

Apply L'Hôpital's Rule repeatedly if necessary, differentiating numerator and denominator until the limit can be evaluated or the behavior is clear.

Since the denominator \(e^{x}\) grows faster than any polynomial, the limit of \(\frac{f(x)}{g(x)}\) as \(x \to \infty\) will approach zero, indicating that \(f\) grows slower than \(g\).

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Key Concepts

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

Growth Rates of Functions

Understanding how functions grow as x approaches infinity involves comparing their dominant terms. Polynomial functions like 10x³ grow at a rate determined by the highest power of x, while exponential functions like eˣ grow much faster, eventually outpacing any polynomial.

Limits at Infinity

Evaluating the limit of the ratio f(x)/g(x) as x approaches infinity helps determine relative growth rates. If the limit is zero, f grows slower; if infinite, f grows faster; if finite and nonzero, they grow at comparable rates.

Exponential vs Polynomial Growth

Exponential functions increase faster than any polynomial function as x becomes very large. This is because exponentials multiply by a constant factor repeatedly, while polynomials increase by powers of x, making exponentials dominant in the long run.

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