Textbook Question
Evaluate the integrals in Exercises 77–90.
81. ∫dy/(y²-2y+5)
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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 7.5.3
In Exercises 1–6, use l’Hôpital’s Rule to evaluate the limit. Then evaluate the limit using a method studied in Chapter 2.
3. lim (x → ∞) (5x² - 3x) / (7x² + 1)
Verified step by step guidance1
Identify the limit expression: \(\lim_{x \to \infty} \frac{5x^{2} - 3x}{7x^{2} + 1}\).
Check if the limit is an indeterminate form of type \(\frac{\infty}{\infty}\) by analyzing the degrees of the numerator and denominator as \(x\) approaches infinity.
Apply l'Hôpital's Rule by differentiating the numerator and denominator separately: differentiate \(5x^{2} - 3x\) to get \(10x - 3\), and differentiate \(7x^{2} + 1\) to get \$14x$.
Rewrite the limit using the derivatives: \(\lim_{x \to \infty} \frac{10x - 3}{14x}\), and analyze this new limit.
Alternatively, use algebraic simplification by dividing numerator and denominator by \(x^{2}\) (the highest power of \(x\) in the denominator) to simplify the expression and then evaluate the limit as \(x\) approaches infinity.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Limits at Infinity
Limits at infinity describe the behavior of a function as the input grows without bound. Understanding how polynomials behave for very large values of x helps determine the end behavior of rational functions, often simplifying the evaluation of limits as x approaches infinity.
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Cases Where Limits Do Not Exist
l’Hôpital’s Rule
l’Hôpital’s Rule is a method for evaluating limits that result in indeterminate forms like 0/0 or ∞/∞. It involves differentiating the numerator and denominator separately and then taking the limit of their quotient, which often simplifies the evaluation.
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5:50Power Rules
Algebraic Simplification of Rational Functions
Algebraic simplification involves dividing numerator and denominator by the highest power of x present to simplify the expression. This method, studied in earlier calculus chapters, helps evaluate limits by reducing complex rational functions to simpler forms without using derivatives.
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Related Practice
Textbook Question
In Exercises 25–36, find the derivative of y with respect to the appropriate variable.
33. y = csch⁻¹(1/2)^θ
Textbook Question
Evaluate the integrals in Exercises 97–110.
97. ∫ 3x^(√3) dx
Textbook Question
In Exercises 59–86, find the derivative of y with respect to the given independent variable.
81. y = log₁₀(e^x)
Textbook Question
Evaluate the integrals in Exercises 41–60.
43. ∫6cosh(x/2 - ln3)dx
Textbook Question
84. Find lim(x→∞) (√(x² + 1) - √x).
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