Ch. 7 - Transcendental Functions

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

Hass 15th Edition

Ch. 7 - Transcendental Functions

Problem 7.8.6e

Chapter 7, Problem 7.8.6e

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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Identify the dominant terms in the function as \(x \to \infty\). The function given is \(f(x) = x - 2\ln(x)\), which consists of a linear term \(x\) and a logarithmic term \(\ln(x)\).

Recall the growth rates of common functions as \(x \to \infty\): polynomial functions like \(x\) grow faster than logarithmic functions like \(\ln(x)\), and constants multiplied by \(\ln(x)\) do not change its growth rate class.

Compare each term separately to \(\ln(x)\): the term \(x\) grows faster than \(\ln(x)\), while the term \(-2\ln(x)\) grows at the same rate as \(\ln(x)\) but with a negative coefficient.

Since \(x\) dominates \(-2\ln(x)\) for large \(x\), the overall function \(f(x) = x - 2\ln(x)\) grows faster than \(\ln(x)\) as \(x \to \infty\).

Conclude that \(f(x)\) grows faster than \(\ln(x)\), it does not grow at the same rate, nor does it grow slower.

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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 as x Approaches Infinity

Understanding how functions behave as x approaches infinity is essential for comparing their growth rates. Functions can grow faster, slower, or at the same rate depending on their dominant terms. For example, polynomial functions generally grow faster than logarithmic functions, while logarithmic functions grow slower than linear functions.

Asymptotic Comparison Using Limits

To compare growth rates, we often use limits of ratios of functions as x approaches infinity. If the limit of f(x)/g(x) is zero, f grows slower than g; if it is infinity, f grows faster; if it is a finite nonzero constant, they grow at the same rate. This method helps rigorously classify growth behavior.

Dominant Terms in Composite Functions

When functions combine multiple terms, the term with the highest growth rate dominates the overall behavior as x→∞. For example, in x - 2ln(x), the linear term x grows faster than the logarithmic term ln(x), so the function's growth rate is primarily determined by x.

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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

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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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)

Verified video answer for a similar problem:

Key Concepts

Growth Rates of Functions as x Approaches Infinity

Asymptotic Comparison Using Limits

Dominant Terms in Composite Functions

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)