Domanda del libro di testo
Evaluate the integrals in Exercises 77–90.
87. ∫(x²+2x-1)/(x²+9) dx
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 7.5.12
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
12. lim (x → ∞) (x - 8x²) / (12x² + 5x)
Guida verificata passo dopo passo1
Identify the limit expression: \(\lim_{x \to \infty} \frac{x - 8x^{2}}{12x^{2} + 5x}\).
Check the form of the limit by analyzing the degrees of the numerator and denominator as \(x\) approaches infinity. Both numerator and denominator tend to infinity, so the limit is of the form \(\frac{\infty}{\infty}\), which is an indeterminate form suitable for l'Hôpital's Rule.
Apply l'Hôpital's Rule by differentiating the numerator and denominator separately with respect to \(x\): differentiate numerator \(\frac{d}{dx}(x - 8x^{2})\) and denominator \(\frac{d}{dx}(12x^{2} + 5x)\).
Write the new limit expression using the derivatives: \(\lim_{x \to \infty} \frac{\frac{d}{dx}(x - 8x^{2})}{\frac{d}{dx}(12x^{2} + 5x)}\).
Evaluate the new limit by simplifying the derivatives and then analyzing the behavior as \(x\) approaches infinity to find the limit.
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Limits at Infinity
Limits at infinity describe the behavior of a function as the input grows without bound. Understanding how to evaluate these limits helps determine the end behavior of rational functions, often by comparing the degrees of polynomials in the numerator and denominator.
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Cases Where Limits Do Not Exist
Indeterminate Forms
Indeterminate forms like ∞/∞ or 0/0 occur when direct substitution in a limit does not yield a clear answer. Recognizing these forms is essential because they signal the need for techniques like l’Hôpital’s rule to evaluate the limit properly.
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Percorso guidato
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l’Hôpital’s Rule
l’Hôpital’s rule provides a method to evaluate limits that result in indeterminate forms by differentiating the numerator and denominator separately. Applying this rule simplifies the limit calculation, especially for rational functions where direct substitution is inconclusive.
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