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
Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 7, Problem 7.8.9.e
9. True, or false? As x→∞,
e. e^x = o(e^(2x))
Verified step by step guidance1
Recall the definition of the little-o notation: For functions \(f(x)\) and \(g(x)\), we say \(f(x) = o(g(x))\) as \(x \to \infty\) if \(\lim_{x \to \infty} \frac{f(x)}{g(x)} = 0\).
Identify the functions in the problem: \(f(x) = e^{x}\) and \(g(x) = e^{2x}\).
Set up the limit to check the little-o relationship: \(\lim_{x \to \infty} \frac{e^{x}}{e^{2x}}\).
Simplify the fraction inside the limit: \(\frac{e^{x}}{e^{2x}} = e^{x - 2x} = e^{-x}\).
Evaluate the limit: \(\lim_{x \to \infty} e^{-x}\). Since \(e^{-x} = \frac{1}{e^{x}}\) and \(e^{x} \to \infty\), the limit is \(0\), confirming that \(e^{x} = o(e^{2x})\) as \(x \to \infty\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Big-O and Little-o Notation
Big-O and little-o notation describe the limiting behavior of functions as the input grows large. Little-o notation, f(x) = o(g(x)), means f(x) grows much slower than g(x), so the ratio f(x)/g(x) approaches zero as x approaches infinity.
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Exponential Growth Rates
Exponential functions like e^x and e^(2x) grow rapidly, but e^(2x) grows faster than e^x because its exponent is larger. Comparing their growth rates helps determine which function dominates as x approaches infinity.
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Limits at Infinity
Evaluating limits as x approaches infinity involves analyzing how functions behave for very large x. For example, the limit of the ratio e^x / e^(2x) simplifies to e^{-x}, which approaches zero, indicating relative growth rates.
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Related Practice
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?
f. (e^x)/2
Textbook Question
112. True, or false? Give reasons for your answers.
e. sec^(-1)x = O(1)
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
2. Express the following logarithms in terms of ln 5 and ln 7.
f. (ln35 + ln(1/7))/(ln25)
Textbook Question
1. Express the following logarithms in terms of ln 2 and ln 3.
f. ln √(13.5)