Textbook Question
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)
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 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
Chapter 2, Problem 2.2.49
Limits with trigonometric functions
Find the limits in Exercises 43–50.
limx→−π √(x + 4) cos(x + π)
Verified step by step guidance1
First, understand the problem: We need to find the limit of the function \( \sqrt{x + 4} \cos(x + \pi) \) as \( x \) approaches \( -\pi \).
Substitute \( x = -\pi \) into the expression \( \sqrt{x + 4} \). This gives \( \sqrt{-\pi + 4} \). Calculate \( -\pi + 4 \) to find the value inside the square root.
Next, substitute \( x = -\pi \) into the expression \( \cos(x + \pi) \). This simplifies to \( \cos(-\pi + \pi) = \cos(0) \). Recall that \( \cos(0) = 1 \).
Combine the results from the previous steps: The limit expression becomes \( \sqrt{-\pi + 4} \times 1 \).
Evaluate \( \sqrt{-\pi + 4} \) to find the final limit value. Ensure that the expression inside the square root is non-negative, as the square root function is defined for non-negative values.
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Video duration:
1mKey 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 are essential for understanding continuity, derivatives, and integrals. In this context, evaluating the limit as x approaches -π involves determining the behavior of the function near that point.
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05:50
One-Sided Limits
Trigonometric Functions
Trigonometric functions, such as cosine, are periodic functions that relate angles to ratios of sides in right triangles. They play a crucial role in calculus, especially when dealing with limits, derivatives, and integrals involving angles. Understanding how these functions behave near specific points is key to solving limit problems.
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6:04Introduction to Trigonometric Functions
Square Root Function
The square root function, denoted as √(x), is defined for non-negative values of x and represents the principal square root. In limit problems, it is important to consider the domain of the function and how it behaves as the input approaches certain values. In this case, √(x + 4) must be evaluated as x approaches -π to ensure the expression remains valid.
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Related Practice
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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Textbook Question
Finding Limits of Differences When x → ±∞
Find the limits in Exercises 84–90. (Hint: Try multiplying and dividing by the conjugate.)
lim x → ∞ (√(x + 9) − √(x + 4))
Textbook Question
Using the Formal Definition
Prove the limit statements in Exercises 37–50.
limx→1 f(x) = 1 if f(x) = {x², x ≠ 1
2, x = 1
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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