Textbook Question
Using limθ→0 sin θ / θ = 1
Find the limits in Exercises 23–46.
limh→0− h / sin 3h
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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 2.6.69
Domains and Asymptotes
Determine the domain of each function in Exercises 69–72. Then use various limits to find the asymptotes.
y = 4 + 3x² / (x² + 1)
Verified step by step guidance1
Step 1: Identify the domain of the function. The function given is \( y = 4 + \frac{3x^2}{x^2 + 1} \). Since the denominator \( x^2 + 1 \) is always positive and never zero for all real numbers, the domain of the function is all real numbers, \( (-\infty, \infty) \).
Step 2: Determine the horizontal asymptote by evaluating the limit of the function as \( x \to \infty \) and \( x \to -\infty \). For large values of \( x \), the term \( \frac{3x^2}{x^2 + 1} \) approaches \( \frac{3x^2}{x^2} = 3 \). Therefore, the horizontal asymptote is \( y = 4 + 3 = 7 \).
Step 3: Check for vertical asymptotes by finding values of \( x \) that make the denominator zero. Since \( x^2 + 1 \) is never zero for real \( x \), there are no vertical asymptotes.
Step 4: Consider the behavior of the function as \( x \to 0 \). Substitute \( x = 0 \) into the function to find \( y = 4 + \frac{3(0)^2}{0^2 + 1} = 4 \). This confirms that there is no vertical asymptote at \( x = 0 \).
Step 5: Summarize the findings. The domain of the function is all real numbers, \( (-\infty, \infty) \). The function has a horizontal asymptote at \( y = 7 \) and no vertical asymptotes.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Domain of a Function
The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. For rational functions, the domain is typically restricted by values that make the denominator zero, as these would lead to undefined outputs. In the given function, identifying the domain involves analyzing the expression to determine any restrictions on x.
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Limits
Limits are a fundamental concept in calculus that describe the behavior of a function as the input approaches a certain value. They are essential for understanding continuity, derivatives, and asymptotic behavior. In the context of finding asymptotes, limits help determine the values that the function approaches as x approaches infinity or any critical points where the function may not be defined.
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Asymptotes
Asymptotes are lines that a graph approaches but never touches or crosses. There are three types: vertical asymptotes, which occur where the function is undefined (often due to division by zero); horizontal asymptotes, which describe the behavior of the function as x approaches infinity; and oblique asymptotes, which occur in certain rational functions. Identifying asymptotes involves using limits to analyze the function's behavior at critical points.
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Related Practice
Textbook Question
Horizontal and Vertical Asymptotes
Determine the domain and range of y = (√16―x²) / (x―2).
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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) = (x + 1)/(x² + 3)
Textbook Question
At what points are the functions in Exercises 13–30 continuous?
f(x) = { (x³ − 8)/(x² − 4), x ≠ 2, x ≠ −2
3, x = 2
4, x = −2
Textbook Question
Limits of Average Rates of Change
Because of their connection with secant lines, tangents, and instantaneous rates, limits of the form limh→0 (f(x+h) − f(x)) / h occur frequently in calculus. In Exercises 57–62, evaluate this limit for the given value of x and function f.
f(x) = x², x = 1
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Textbook Question
Limits with trigonometric functions
Find the limits in Exercises 43–50.
limx→−π √(x + 4) cos(x + π)