Domanda del libro di testo
82. Use the definitions of the hyperbolic functions to find each of the following limits.
f. lim(x→∞) coth x
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Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- 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 Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 7, Problema 7.8.4e
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
Guida verificata passo dopo passo1
Identify the dominant term in the function \(x^3 - x^2\) as \(x \to \infty\). The dominant term is the one with the highest power of \(x\), which is \(x^3\) in this case.
Recall that the growth rate of a function as \(x \to \infty\) is determined by its highest power term. Here, \(x^3\) grows faster than \(x^2\) because the exponent 3 is greater than 2.
Compare the dominant term \(x^3\) with \(x^2\): since \(x^3\) grows faster than \(x^2\), the function \(x^3 - x^2\) grows faster than \(x^2\) as \(x \to \infty\).
To confirm, consider the limit \(\lim_{x \to \infty} \frac{x^3 - x^2}{x^2} = \lim_{x \to \infty} (x - 1) = \infty\), which shows the function grows faster than \(x^2\).
Therefore, \(x^3 - x^2\) grows faster than \(x^2\) as \(x\) approaches infinity.
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Asymptotic Growth Rates
Asymptotic growth rates describe how functions behave as the input variable approaches infinity. Comparing growth rates helps determine which functions increase faster, slower, or at the same rate by focusing on dominant terms and ignoring lower-order terms.
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In polynomial functions, the term with the highest exponent dominates the function's behavior for large values of x. For example, in x³ - x², the x³ term grows faster than x², so the overall function behaves like x³ as x approaches infinity.
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