Textbook Question
Infinite Limits
Find the limits in Exercises 37–48. Write ∞ or −∞ where appropriate.
lim x→0⁺ 1 / 3x
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 2.3.44
Using the Formal Definition
Prove the limit statements in Exercises 37–50.
limx → √3 1/x² = 1/3
Verified step by step guidance1
Understand the formal definition of a limit: For a function f(x) and a limit L, we say that lim(x → a) f(x) = L if for every ε > 0, there exists a δ > 0 such that 0 < |x - a| < δ implies |f(x) - L| < ε.
Identify the function and the limit in the problem: Here, f(x) = 1/x² and L = 1/3, with a = √3.
Set up the inequality based on the definition: We need to show that for every ε > 0, there exists a δ > 0 such that 0 < |x - √3| < δ implies |1/x² - 1/3| < ε.
Simplify the inequality |1/x² - 1/3| < ε: This can be rewritten as |(3 - x²)/(3x²)| < ε. Focus on finding a δ that satisfies this condition.
Choose δ appropriately: Analyze the expression |3 - x²| < 3εx² and find a δ that works for the given ε. Consider the behavior of the function around x = √3 to ensure the inequality holds.
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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 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
Continuous Functions
A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. Understanding continuity helps in evaluating limits, especially when dealing with rational functions like 1/x², as it ensures that the function behaves predictably near the limit point.
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05:34Intro to Continuity
Rational Functions
Rational functions are ratios of polynomials, and their limits can often be evaluated by direct substitution unless the function is undefined at that point. In this case, analyzing the behavior of 1/x² as x approaches √3 is essential for determining the limit, which simplifies to 1/(√3)² = 1/3.
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6:04Intro to Rational Functions
Related Practice
Textbook Question
Finding One-Sided Limits Algebraically
Find the limits in Exercises 11–20.
limx→1− (1/(x + 1))((x + 6)/x)((3 − x)/7)
Textbook Question
Using the Formal Definition
Prove the limit statements in Exercises 37–50.
lim x→0 √(4 − x) = 2
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 → −∞ (√(x² + 3) + x)
Textbook Question
Limits and Infinity
Find the limits in Exercises 37–46.
x²/³ + x⁻¹
lim --------------------
x→∞ x²/³ + cos²x
Textbook Question
Finding Limits
In Exercises 3–8, find the limit of each function (a) as x → ∞ and (b) as x → −∞. (You may wish to visualize your answer with a graphing calculator or computer.)
f(x) = 2/x − 3