Domanda del libro di testo
Evaluate the integrals in Exercises 31–78.
73. ∫dx/√(-2x-x²)
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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 7.5.9
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
9. lim (t → -3) (t³ - 4t + 15) / (t² - t - 12)
Guida verificata passo dopo passo1
First, identify the limit expression: \(\lim_{t \to -3} \frac{t^{3} - 4t + 15}{t^{2} - t - 12}\).
Evaluate the numerator and denominator separately at \(t = -3\) to check if the limit is an indeterminate form. Calculate \((-3)^{3} - 4(-3) + 15\) and \((-3)^{2} - (-3) - 12\).
If both numerator and denominator evaluate to 0, then the limit is of the form \(\frac{0}{0}\), and l’Hôpital’s Rule can be applied.
Apply l’Hôpital’s Rule by differentiating the numerator and denominator separately with respect to \(t\): find \(\frac{d}{dt}(t^{3} - 4t + 15)\) and \(\frac{d}{dt}(t^{2} - t - 12)\).
After differentiation, substitute \(t = -3\) into the new expression \(\frac{\frac{d}{dt}(\text{numerator})}{\frac{d}{dt}(\text{denominator})}\) to find the limit.
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Limits and Indeterminate Forms
Limits describe the behavior of a function as the input approaches a particular value. When direct substitution results in an indeterminate form like 0/0 or ∞/∞, special techniques such as l’Hôpital’s rule are needed to evaluate the limit.
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One-Sided Limits
l’Hôpital’s Rule
l’Hôpital’s rule provides a method to evaluate limits that yield indeterminate forms by differentiating the numerator and denominator separately and then taking the limit of their quotient. It applies only when the original limit results in 0/0 or ∞/∞.
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5:50Power Rules
Differentiation of Polynomial Functions
Differentiation involves finding the derivative of a function, which represents its rate of change. For polynomials, derivatives are found by applying the power rule term-by-term, which is essential when using l’Hôpital’s rule to differentiate numerator and denominator.
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07:00Taylor Polynomials
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