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Ch. 7 - Transcendental Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 7 - Transcendental FunctionsProblem 7.8.2e
Chapter 7, Problem 7.8.2e

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)

Verified step by step guidance
1
Recall that the function \(e^x\) grows exponentially as \(x \to \infty\), meaning it increases very rapidly without bound.
Consider the given function \(e^{-x}\). Rewrite it as \(\frac{1}{e^x}\) to better understand its behavior as \(x \to \infty\).
As \(x \to \infty\), \(e^x\) becomes very large, so \(\frac{1}{e^x}\) approaches 0. This means \(e^{-x}\) approaches 0 and does not grow; it actually decays to zero.
Compare the growth rates: since \(e^{-x}\) approaches zero while \(e^x\) grows without bound, \(e^{-x}\) grows slower than \(e^x\) as \(x \to \infty\).
Therefore, \(e^{-x}\) grows slower than \(e^x\) 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.

Exponential Growth and Decay

Exponential functions like e^x grow rapidly as x approaches infinity, while functions like e^(-x) decay towards zero. Understanding the difference between growth and decay is essential to compare their rates as x→∞.
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Exponential Growth & Decay

Limits and Asymptotic Behavior

Analyzing the limit of a function as x approaches infinity helps determine its growth rate relative to others. Functions that approach infinity faster have higher growth rates, while those approaching zero or finite values grow slower.
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Asymptotes of Hyperbolas

Comparing Growth Rates of Functions

Functions can be compared by examining their dominant terms as x→∞. For example, polynomial, exponential, and logarithmic functions grow at different rates, with exponential functions like e^x generally outpacing polynomials and logarithms.
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Related Practice
Textbook Question

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

e. x^3 - x^2

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:

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

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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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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