Domanda del libro di testo
Use l’Hôpital’s Rule to find the limits in Exercises 85–108.
93. lim(x→0) (csc(x) - cot(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.PE.109b
109. Does f grow faster, slower, or at the same rate as g as x→∞? Give reasons for your answers.
b. f(x)=x, g(x)=x + 1/x
Guida verificata passo dopo passo1
Identify the given functions: \(f(x) = x\) and \(g(x) = x + \frac{1}{x}\).
As \(x \to \infty\), analyze the behavior of each function. For \(f(x) = x\), the function grows without bound linearly.
For \(g(x) = x + \frac{1}{x}\), note that \(\frac{1}{x} \to 0\) as \(x \to \infty\), so \(g(x)\) behaves like \(x\) plus a very small positive term.
Compare the growth rates by considering the ratio \(\frac{f(x)}{g(x)} = \frac{x}{x + \frac{1}{x}}\). Simplify this expression to understand the limit as \(x \to \infty\).
Evaluate the limit \(\lim_{x \to \infty} \frac{f(x)}{g(x)}\) to determine if it approaches 0, a finite nonzero constant, or infinity, which tells us if \(f\) grows slower, at the same rate, or faster than \(g\) respectively.
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Asymptotic Behavior of Functions
Asymptotic behavior describes how functions behave as the input grows very large (x→∞). It helps compare growth rates by focusing on dominant terms and ignoring smaller ones that become insignificant at infinity.
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Percorso guidato
5:50
Asymptotes of Hyperbolas
Limits and Limit Comparison
Limits evaluate the value a function approaches as x approaches infinity. Comparing the limit of the ratio f(x)/g(x) as x→∞ determines if one function grows faster, slower, or at the same rate as the other.
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07:45Limit Comparison Test
Dominant Terms in Functions
The dominant term in a function is the part that grows fastest as x→∞. For example, in g(x) = x + 1/x, the term x dominates because 1/x approaches zero, so the growth rate is mainly determined by x.
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05:44Divergence Test (nth Term Test)
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