Textbook Question
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))
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 2.4.46
Using limθ→0 sin θ / θ = 1
Find the limits in Exercises 23–46.
limx→0 (cos²x − cos x) / x²
Verified step by step guidance1
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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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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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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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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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Related Practice
Textbook Question
Define g(4) in a way that extends g(x) = (x² − 16)/(x² − 3x − 4) to be continuous at x = 4.
Textbook Question
Using limθ→0 sin θ / θ = 1
Find the limits in Exercises 23–46.
limθ→0 tan θ / θ²cot 3θ
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Textbook Question
Limits of quotients
Find the limits in Exercises 23–42.
limx→−5 (x² + 3x − 10) / x + 5
Textbook Question
Explain why the equation cos x = x has at least one solution.
Textbook Question
Suppose that a function f(x) is defined for all real values of x except x=c. Can anything be said about the existence of limx→c f(x)? Give reasons for your answer.