Textbook Question
Define g(4) in a way that extends g(x) = (x² − 16)/(x² − 3x − 4) to be continuous at x = 4.
Ch. 2 - Limits and Continuity
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 2, Problem 2.2.25
Limits of quotients
Find the limits in Exercises 23–42.
limx→−5 (x² + 3x − 10) / x + 5
Verified step by step guidance1
First, identify the type of limit problem. This is a limit of a quotient as x approaches -5.
Check if direct substitution of x = -5 into the expression (x² + 3x − 10) / (x + 5) results in an indeterminate form like 0/0.
Substitute x = -5 into the numerator: x² + 3x − 10 becomes (-5)² + 3(-5) − 10.
Substitute x = -5 into the denominator: x + 5 becomes -5 + 5.
If the substitution results in 0/0, factor the numerator x² + 3x − 10 to see if it can be simplified with the denominator x + 5. Look for common factors to cancel out and resolve the indeterminate form.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Limits
Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. They help in understanding the behavior of functions near specific points, especially when direct substitution may lead to indeterminate forms. In this case, evaluating the limit as x approaches -5 requires analyzing the function's behavior around that point.
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One-Sided Limits
Quotient of Functions
The quotient of functions involves dividing one function by another. When finding limits of quotients, it is essential to consider the behavior of both the numerator and denominator as the variable approaches a specific value. If the denominator approaches zero while the numerator does not, the limit may be undefined or infinite, necessitating further analysis or simplification.
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06:43The Quotient Rule
Factoring and Simplifying
Factoring and simplifying expressions is a crucial technique in calculus, especially when dealing with limits. By factoring the numerator and denominator, one can often cancel common terms, which can resolve indeterminate forms like 0/0. In this problem, factoring the numerator will help in simplifying the expression before evaluating the limit as x approaches -5.
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6:36Simplifying Trig Expressions
Related Practice
Textbook Question
Using limθ→0 sin θ / θ = 1
Find the limits in Exercises 23–46.
limθ→0 tan θ / θ²cot 3θ
1
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Textbook Question
Using limθ→0 sin θ / θ = 1
Find the limits in Exercises 23–46.
limx→0 (cos²x − cos x) / x²
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Textbook Question
Explain why the equation cos x = x has at least one solution.
Textbook Question
Suppose that a function f(x) is defined for all real values of x except x=c. Can anything be said about the existence of limx→c f(x)? Give reasons for your answer.
Textbook Question
Limits of Rational Functions
In Exercises 13–22, find the limit of each rational function (a) as x → ∞ and (b) as x → −∞. Write ∞ or −∞ where appropriate.
f(x) = (2x + 3)/(5x + 7)