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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 7.8.1e
Chapter 7, Problem 7.8.1e

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

Verified step by step guidance
1
Recall that the growth rate of exponential functions as \(x \to \infty\) depends on the base of the exponent. The function \(e^x\) has base \(e \approx 2.718\).
Compare the base of the given function \((\frac{3}{2})^x\) with the base \(e\) of \(e^x\). Here, \(\frac{3}{2} = 1.5\) which is less than \(e\).
Since \(1.5 < e\), the function \((\frac{3}{2})^x\) grows slower than \(e^x\) as \(x \to \infty\).
To summarize, if the base of the exponential function is less than \(e\), it grows slower than \(e^x\); if it equals \(e\), it grows at the same rate; and if it is greater than \(e\), it grows faster.
Therefore, \((\frac{3}{2})^x\) 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 Base Comparison

Exponential functions have the form a^x, where a > 0. The growth rate depends on the base a: if a > 1, the function grows exponentially. Comparing bases helps determine which function grows faster as x approaches infinity.
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Exponential Growth & Decay

Limit Behavior as x Approaches Infinity

Analyzing the limit of the ratio of two functions as x→∞ reveals their relative growth rates. If the limit is zero, the numerator grows slower; if infinite, it grows faster; if finite and nonzero, they grow at the same rate.
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Growth Rate Comparison of Exponential Functions

For exponential functions e^x and a^x, growth rate depends on the base: e ≈ 2.718. If a < e, a^x grows slower than e^x; if a = e, they grow at the same rate; if a > e, a^x grows faster than e^x as x→∞.
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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

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.


70. y= x³/(x²+1), -1 ≤ x ≤ 1, x_0=1/2

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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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