Domanda del libro di testo
Using limθ→0 sin θ / θ = 1
Find the limits in Exercises 23–46.
limh→0− h / sin 3h
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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 2.4.36
Using limθ→0 sin θ / θ = 1
Find the limits in Exercises 23–46.
limh→0 sin(sin h) / sin h
Guida verificata passo dopo passo1
First, recognize that the problem involves finding the limit of the expression lim(h→0) sin(sin(h)) / sin(h). This is a limit problem that can be approached using the known limit lim(θ→0) sin(θ) / θ = 1.
To solve this, consider the substitution u = sin(h). As h approaches 0, u also approaches 0 because sin(h) is continuous and sin(0) = 0.
Rewrite the original limit in terms of u: lim(u→0) sin(u) / u. This transformation is valid because as h approaches 0, u approaches 0.
Now, apply the known limit property: lim(θ→0) sin(θ) / θ = 1. Since u is approaching 0, this property can be directly applied to the expression lim(u→0) sin(u) / u.
Conclude that the limit of the original expression lim(h→0) sin(sin(h)) / sin(h) is equal to 1, based on the substitution and the known limit property.
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Limit of a Function
The limit of a function describes the behavior of the function as the input approaches a particular value. In calculus, understanding limits is crucial for analyzing the continuity and differentiability of functions. For the given problem, evaluating the limit as h approaches 0 is essential to determine the behavior of the function sin(sin h) / sin h.
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Limits of Rational Functions: Denominator = 0
Squeeze Theorem
The Squeeze Theorem is a method used to find the limit of a function by 'squeezing' it between two other functions whose limits are known and equal. This theorem is particularly useful when direct substitution in a limit results in an indeterminate form. In this problem, the Squeeze Theorem can help confirm the limit of sin(sin h) / sin h as h approaches 0.
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Percorso guidato
06:11Fundamental Theorem of Calculus Part 1
Limit of sin(θ)/θ as θ approaches 0
The limit limθ→0 sin(θ)/θ = 1 is a fundamental result in calculus, often used to evaluate limits involving trigonometric functions. This limit is derived from the unit circle and the small-angle approximation. In the given problem, this concept is directly applied to simplify and solve the limit of sin(sin h) / sin h as h approaches 0.
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