Domanda del libro di testo
Domains and Asymptotes
Determine the domain of each function in Exercises 69–72. Then use various limits to find the asymptotes.
y = x³ / (x³ − 8)
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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 2.4.43
Using limθ→0 sin θ / θ = 1
Find the limits in Exercises 23–46.
limθ→0 tan θ / θ²cot 3θ
Guida verificata passo dopo passo1
First, recognize that the limit involves trigonometric functions and their behavior as θ approaches 0. We will use the known limit limθ→0 sin θ / θ = 1 to help simplify the expression.
Rewrite tan θ in terms of sin θ and cos θ: tan θ = sin θ / cos θ. Similarly, rewrite cot 3θ as cos 3θ / sin 3θ.
Substitute these expressions into the limit: limθ→0 (sin θ / cos θ) / (θ² * (cos 3θ / sin 3θ)).
Simplify the expression: limθ→0 (sin θ * sin 3θ) / (θ² * cos θ * cos 3θ).
Apply the limit properties and the known limit limθ→0 sin θ / θ = 1 to evaluate the limit as θ approaches 0. Consider the behavior of each trigonometric function and their derivatives at θ = 0.
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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 θ approaches 0 is essential to determine the behavior of the expression tan θ / θ²cot 3θ.
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Limits of Rational Functions: Denominator = 0
Trigonometric Limits
Trigonometric limits involve evaluating limits that include trigonometric functions like sine, cosine, and tangent. A fundamental trigonometric limit is limθ→0 sin θ / θ = 1, which is often used to simplify expressions involving small angles. This concept is key in solving the given problem, as it helps in simplifying the trigonometric components of the expression.
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Percorso guidato
6:04Introduction to Trigonometric Functions
L'Hôpital's Rule
L'Hôpital's Rule is a method for finding limits of indeterminate forms like 0/0 or ∞/∞. It states that if the limit of f(x)/g(x) as x approaches a value results in an indeterminate form, the limit can be found by differentiating the numerator and denominator separately. This rule is useful in the given problem if direct substitution leads to an indeterminate form, allowing for further simplification.
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5:50Power Rules
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