Ch. 7 - Transcendental Functions

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 7 - Transcendental Functions

Problem 7.8.5e

Chapter 7, Problem 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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Key Concepts

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

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.

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.

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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Related Practice

Textbook Question

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)

Textbook Question

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:

d. Find the equation for the tangent line to g at the point (f(x_0), x_0) located symmetrically across the 45° line y=x (which is the graph of the identity function). Use Theorem 1 to find the slope of this tangent line.

68. y= (3x+2)/(2x-11), -2 ≤ x ≤ 2, x_0=1/2

Textbook Question

70. y= x³/(x²+1), -1 ≤ x ≤ 1, x_0=1/2

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Textbook Question

1. 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. (3/2)^x

Textbook Question

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)

Textbook Question

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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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

Verified video answer for a similar problem:

Key Concepts

Growth Rates of Functions

Logarithmic vs. Polynomial Growth

Asymptotic Behavior and Limits

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