Domanda del libro di testo
Find the limits in Exercises 31–40. Are the functions continuous at the point being approached?
lim t → 0 sin (π/2 cos (tan t))
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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 2.4.46
Using limθ→0 sin θ / θ = 1
Find the limits in Exercises 23–46.
limx→0 (cos²x − cos x) / x²
Guida verificata passo dopo passo1
First, recognize that the limit involves a trigonometric function and a polynomial expression. We need to simplify the expression (cos²x − cos x) / x² as x approaches 0.
Use the trigonometric identity: cos²x = 1 - sin²x. Substitute this into the expression to get: ((1 - sin²x) - cos x) / x².
Next, apply the Taylor series expansion for cos x around x = 0: cos x ≈ 1 - x²/2. Substitute this approximation into the expression.
Simplify the expression using the Taylor expansion: ((1 - sin²x) - (1 - x²/2)) / x². This simplifies to: (sin²x - x²/2) / x².
Finally, evaluate the limit as x approaches 0. Consider the behavior of sin²x/x² as x approaches 0, using the fact that limθ→0 sin θ / θ = 1, and simplify the expression further to find the limit.
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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 x approaches 0 is essential to determine the behavior of the expression (cos²x − cos x) / x².
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Limits of Rational Functions: Denominator = 0
L'Hôpital's Rule
L'Hôpital's Rule is a method for finding limits of indeterminate forms, such as 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 particularly useful for the given problem, where direct substitution leads to an indeterminate form.
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5:50Power Rules
Trigonometric Limits
Trigonometric limits involve the behavior of trigonometric functions as the input approaches a specific value. A fundamental trigonometric limit is limθ→0 sin θ / θ = 1, which is often used to evaluate limits involving sine and cosine functions. Understanding this concept helps in simplifying expressions and finding limits involving trigonometric functions, as seen in the given problem.
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Percorso guidato
6:04Introduction to Trigonometric Functions
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