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Infinite Limits
Find the limits in Exercises 37–48. Write ∞ or −∞ where appropriate.
lim x→0⁺ 1 / 3x
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Ch. 2 - Limits and Continuity
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- 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
Capitolo 2, Problema 2.3.50
Using the Formal Definition
Prove the limit statements in Exercises 37–50.
lim x→0 x² sin (1/x) = 0
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Guida verificata passo dopo passo1
Understand the formal definition of a limit: For a function f(x) to have a limit L as x approaches a, for every ε > 0, there exists a δ > 0 such that if 0 < |x - a| < δ, then |f(x) - L| < ε.
Identify the function and the limit: Here, f(x) = x² sin(1/x) and we want to prove that lim x→0 f(x) = 0.
Set up the inequality using the formal definition: We need to show that for every ε > 0, there exists a δ > 0 such that if 0 < |x| < δ, then |x² sin(1/x) - 0| < ε.
Simplify the expression: Notice that |x² sin(1/x)| = |x²| |sin(1/x)|. Since |sin(1/x)| ≤ 1 for all x, we have |x² sin(1/x)| ≤ |x²|.
Choose δ: To ensure |x²| < ε, we can choose δ = √ε. Then, for 0 < |x| < δ, we have |x²| < δ² = ε, which satisfies the condition |x² sin(1/x)| < ε, proving the limit.
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Limit Definition
The formal definition of a limit states that for a function f(x), the limit as x approaches a value c is L if, for every ε > 0, there exists a δ > 0 such that whenever 0 < |x - c| < δ, it follows that |f(x) - L| < ε. This definition is crucial for proving limit statements rigorously.
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05:43
Definition of the Definite Integral
Squeeze Theorem
The Squeeze Theorem is a method used to find limits of functions that are difficult to evaluate directly. It states that if f(x) ≤ g(x) ≤ h(x) for all x near c (except possibly at c), and if the limits of f(x) and h(x) as x approaches c are both L, then the limit of g(x) as x approaches c is also L. This theorem is particularly useful for functions involving oscillatory behavior, like sin(1/x).
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06:11Fundamental Theorem of Calculus Part 1
Behavior of Oscillatory Functions
Oscillatory functions, such as sin(1/x), do not settle at a single value as x approaches 0; instead, they oscillate between -1 and 1. Understanding this behavior is essential when evaluating limits involving such functions, as it allows us to analyze their impact on the overall limit, particularly when multiplied by terms that approach zero, like x² in this case.
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