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)
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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 2.3.40
Using the Formal Definition
Prove the limit statements in Exercises 37–50.
lim x→0 √(4 − x) = 2
Verified step by step guidance1
Start by recalling the formal definition of a limit: For the limit \( \lim_{x \to a} f(x) = L \), for every \( \epsilon > 0 \), there exists a \( \delta > 0 \) such that if \( 0 < |x - a| < \delta \), then \( |f(x) - L| < \epsilon \).
In this problem, we need to prove that \( \lim_{x \to 0} \sqrt{4 - x} = 2 \). This means for every \( \epsilon > 0 \), we need to find a \( \delta > 0 \) such that if \( 0 < |x - 0| < \delta \), then \( |\sqrt{4 - x} - 2| < \epsilon \).
Consider the expression \( |\sqrt{4 - x} - 2| < \epsilon \). To manipulate this, multiply and divide by the conjugate: \( |\sqrt{4 - x} - 2| = \frac{|(\sqrt{4 - x} - 2)(\sqrt{4 - x} + 2)|}{|\sqrt{4 - x} + 2|} = \frac{|4 - x - 4|}{|\sqrt{4 - x} + 2|} = \frac{|x|}{|\sqrt{4 - x} + 2|} \).
To ensure \( \frac{|x|}{|\sqrt{4 - x} + 2|} < \epsilon \), we need \( |x| < \epsilon \cdot |\sqrt{4 - x} + 2| \). Since \( \sqrt{4 - x} \) is close to 2 when \( x \) is near 0, \( |\sqrt{4 - x} + 2| \) is close to 4. We can choose \( \delta \) such that \( |x| < \epsilon \cdot 4 \).
Finally, choose \( \delta = \min(1, 4\epsilon) \) to ensure that \( |x| < \delta \) implies \( |\sqrt{4 - x} - 2| < \epsilon \). This completes the proof using the formal definition of a limit.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Limit Definition
The formal definition of a limit, often called the epsilon-delta definition, states that for a function f(x) to have a limit L as x approaches a value c, for every ε > 0, there exists a δ > 0 such that if 0 < |x - c| < δ, then |f(x) - L| < ε. This definition is crucial for rigorously proving limit statements.
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Definition of the Definite Integral
Square Root Function
The square root function, √(x), is a fundamental mathematical function that returns the principal square root of a non-negative number x. Understanding its behavior, especially near points of interest like x = 0, is essential for analyzing limits involving square roots, as it affects the continuity and differentiability of the function.
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Limit Properties
Limit properties, such as the limit of a sum, product, or composition of functions, help simplify complex limit problems. For instance, knowing that the limit of a constant is the constant itself, and the limit of a function as x approaches a point can be distributed over addition and multiplication, aids in breaking down and solving limit problems efficiently.
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Related Practice
Textbook Question
Using limθ→0 sin θ / θ = 1
Find the limits in Exercises 23–46.
limt→0 2t / tan t
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Textbook Question
Using limθ→0 sin θ / θ = 1
Find the limits in Exercises 23–46.
limθ→0 sin θ cot 2θ
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Textbook Question
At what points are the functions in Exercises 13–30 continuous?
g(x) = { (x² − x – 6)/(x – 3), x ≠ 3
5, x = 3
253
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Textbook Question
Find the limits in Exercises 31–40. Are the functions continuous at the point being approached?
lim x → π/6 √(csc² x + 5√3 tan x)
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Textbook Question
Centering Intervals About a Point
In Exercises 1–6, sketch the interval (a,b), on the x-axis with the point c inside. Then find a value of δ>0 such that a < x < b whenever 0 < |x−c| < δ.
a=4/9, b=4/7, c=1/2
362
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