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Ch. 7 - Transcendental Functions
Hass - Thomas' Calculus 15th Edition
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Tutti i libri di testoHass 15th EditionCh. 7 - Transcendental FunctionsProblema 7.8.5e
Capitolo 7, Problema 7.8.5e

5. Which of the following functions grow faster than ln(x) as x→∞? Which grow at the same rate as ln(x)? Which grow slower?
e. x

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Recall that the growth rate of functions as \(x \to \infty\) can be compared using limits of their ratios. Specifically, for two functions \(f(x)\) and \(g(x)\), if \(\lim_{x \to \infty} \frac{f(x)}{g(x)} = \infty\), then \(f(x)\) grows faster than \(g(x)\). If the limit is a finite nonzero constant, they grow at the same rate. If the limit is 0, \(f(x)\) grows slower than \(g(x)\).
Identify the functions to compare: here, \(f(x) = x\) and \(g(x) = \ln(x)\).
Set up the limit to compare their growth rates: \(\lim_{x \to \infty} \frac{x}{\ln(x)}\).
Analyze the limit: since \(x\) grows without bound much faster than \(\ln(x)\), this limit tends to \(\infty\), indicating that \(x\) grows faster than \(\ln(x)\) as \(x \to \infty\).
Conclude that the function \(x\) grows faster than \(\ln(x)\) as \(x\) approaches infinity.

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Growth Rates of Functions

Growth rates describe how functions behave as their input becomes very large. Comparing growth rates helps determine which functions increase faster, slower, or at the same pace as others, especially as x approaches infinity.
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Logarithmic vs. Polynomial Growth

Logarithmic functions like ln(x) grow very slowly compared to polynomial functions such as x. As x approaches infinity, polynomial functions increase much faster than logarithmic ones, making them dominant in growth comparisons.
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Asymptotic Behavior and Limits

Asymptotic behavior studies the trend of functions as x approaches infinity. Using limits, we can compare the ratio of two functions to determine if one grows faster, slower, or at the same rate as the other.
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Domanda del libro di testo

9. True, or false? As x→∞,

e. e^x = o(e^(2x))

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

e. e^(-x)

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112. True, or false? Give reasons for your answers.

e. sec^(-1)x = O(1)

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In Exercises 67–72, you will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS:

e. Plot the functions f and g, the identity, the two tangent lines, and the line segment joining the points (x_0, f(x_0)) and (f(x_0), x_0). Discuss the symmetries you see across the main diagonal y=x.


72. y= 2-x-x³, -2 ≤ x ≤ 2, x_0 = 3/2

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Domanda del libro di testo

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

e. x ln(x)

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6. Which of the following functions grow faster than ln(x) as x→∞? Which grow at the same rate as ln(x)? Which grow slower?

e. x - 2ln(x)

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