Textbook Question
For what values of a and b is
g(x) = { ax + 2b, x ≤ 0
x² + 3a – b, 0 < x ≤ 2
3x – 5, x > 2
continuous at every x?
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 2.6.95
Use formal definitions to prove the limit statements in Exercises 93–96.
lim x → 3 (−2 / (x − 3)²) = −∞
Verified step by step guidance1
Understand the problem: We need to prove that the limit of the function \(-\frac{2}{(x-3)^2}\) as \(x\) approaches 3 is \(-\infty\). This involves using the formal definition of limits involving infinity.
Recall the formal definition: For a limit to be \(-\infty\) as \(x\) approaches a value \(c\), for every positive number \(M\), there exists a \(\delta > 0\) such that if \(0 < |x - c| < \delta\), then \(f(x) < -M\).
Apply the definition: We need to show that for every \(M > 0\), there exists a \(\delta > 0\) such that if \(0 < |x - 3| < \delta\), then \(-\frac{2}{(x-3)^2} < -M\).
Solve the inequality: Start by solving \(-\frac{2}{(x-3)^2} < -M\). This simplifies to \(\frac{2}{(x-3)^2} > M\). Rearrange to find \(x\) values that satisfy this inequality.
Determine \(\delta\): From the inequality \((x-3)^2 < \frac{2}{M}\), solve for \(x\) to find \(\delta\) such that \(0 < |x - 3| < \delta\) ensures the inequality holds. This will complete the proof using the formal definition.
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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 involves showing that for every ε > 0, there exists a δ > 0 such that if 0 < |x - c| < δ, then |f(x) - L| < ε. In this context, proving a limit involves demonstrating that as x approaches a specific value, the function approaches a particular limit, which can be finite or infinite.
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05:43
Definition of the Definite Integral
Infinite Limits
Infinite limits occur when the value of a function increases or decreases without bound as the input approaches a certain point. In this problem, the limit is negative infinity, indicating that as x approaches 3, the function value decreases indefinitely. Understanding infinite limits requires recognizing how the function behaves near the point of interest.
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05:50One-Sided Limits
Behavior Near Singularities
Singularities are points where a function is not defined or behaves erratically, often leading to infinite limits. In this problem, x = 3 is a singularity for the function −2/(x−3)², as the denominator approaches zero, causing the function to diverge. Analyzing behavior near singularities involves examining how the function's value changes as it approaches these critical points.
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03:07Cases Where Limits Do Not Exist
Related Practice
Textbook Question
Limits as x → ∞ or x → −∞
The process by which we determine limits of rational functions applies equally well to ratios containing noninteger or negative powers of x. Divide numerator and denominator by the highest power of x in the denominator and proceed from there. Find the limits in Exercises 23–36. Write ∞ or −∞ where appropriate.
lim x → ⁻∞ ((1 − x³) / (x² + 7x))⁵
Textbook Question
Using the Formal Definition
Prove the limit statements in Exercises 37–50.
limx→9 √(x − 5) = 2
Textbook Question
The sign-preserving property of continuous functions Let f be defined on an interval (a, b) and suppose that f(c) ≠ 0 at some c where f is continuous. Show that there is an interval (c − δ, c + δ) about c where f has the same sign as f(c).
Textbook Question
Suppose that f(x) and g(x) are polynomials in x. Can the graph of f(x)/g(x) have an asymptote if g(x) is never zero? Give reasons for your answer.
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views
Textbook Question
Limits as x → ∞ or x → −∞
The process by which we determine limits of rational functions applies equally well to ratios containing noninteger or negative powers of x. Divide numerator and denominator by the highest power of x in the denominator and proceed from there. Find the limits in Exercises 23–36. Write ∞ or −∞ where appropriate.
lim x → ∞ √((8x² − 3) / (2x² + x))