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?
g. x^3 e^(-x)
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 7.8.2g
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?
g. e^(cos(x))
Verified step by step guidance1
Recall that the growth rate of a function as \(x \to \infty\) is determined by its dominant behavior for large \(x\). The function \(e^x\) grows exponentially and very rapidly as \(x\) increases.
Analyze the given function \(g(x) = e^{\cos(x)}\). Since \(\cos(x)\) oscillates between \(-1\) and \(1\), the exponent \(\cos(x)\) does not grow without bound; it remains bounded.
Because \(\cos(x)\) is bounded, \(e^{\cos(x)}\) oscillates between \(e^{-1}\) and \(e^{1}\), which are constant values. This means \(g(x)\) remains bounded and does not increase without limit as \(x \to \infty\).
Compare this behavior to \(e^x\), which grows without bound. Since \(g(x)\) remains bounded, it grows much slower than \(e^x\) as \(x \to \infty\).
Therefore, \(g(x) = e^{\cos(x)}\) grows slower than \(e^x\) 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.
Exponential Growth and Limits
Exponential growth describes functions like e^x that increase rapidly as x approaches infinity. Understanding limits helps compare how fast functions grow by analyzing their behavior as x→∞, determining if one function outpaces, matches, or lags behind another.
Recommended video:
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Exponential Growth & Decay
Behavior of Composite Functions
Composite functions like e^(cos(x)) combine an exponential with a bounded function (cos(x)). Since cos(x) oscillates between -1 and 1, e^(cos(x)) oscillates between e^(-1) and e^1, remaining bounded and not growing without limit as x→∞.
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3:48Evaluate Composite Functions - Special Cases
Growth Rate Comparison
Comparing growth rates involves analyzing dominant terms and their limits. A function that remains bounded or oscillates does not grow faster than an unbounded exponential like e^x. Functions with the same dominant exponential term grow at the same rate, while bounded functions grow slower.
Recommended video:
04:16Intro To Related Rates
Related Practice
Textbook Question
82. Use the definitions of the hyperbolic functions to find each of the following limits.
f. lim(x→∞) coth x
1
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Textbook Question
9. True, or false? As x→∞,
g. ln(x) = o(ln(2x))
1
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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?
f. (e^x)/2
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?
g. 1/x
Textbook Question
1. Express the following logarithms in terms of ln 2 and ln 3.
f. ln √(13.5)