교과서 질문
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))
237
views
Ch. 2 - Limits and Continuity
Hass15th EditionThomas' CalculusISBN: 9780137616077
Not the one you use?Change textbookSelect a different textbook
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 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
2장, 문제 2.4.46
Using limθ→0 sin θ / θ = 1
Find the limits in Exercises 23–46.
limx→0 (cos²x − cos x) / x²
검증된 단계별 안내1
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.
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
5m도움이 되었나요?
주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
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.
추천 영상:
06:11
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.
추천 영상:
가이드 코스
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.
추천 영상:
6:04Introduction to Trigonometric Functions
관련 실천
교과서 질문
Define g(4) in a way that extends g(x) = (x² − 16)/(x² − 3x − 4) to be continuous at x = 4.
246
views
교과서 질문
Using limθ→0 sin θ / θ = 1
Find the limits in Exercises 23–46.
limθ→0 tan θ / θ²cot 3θ
289
views
교과서 질문
Limits of quotients
Find the limits in Exercises 23–42.
limx→−5 (x² + 3x − 10) / x + 5
175
views
교과서 질문
Explain why the equation cos x = x has at least one solution.
307
views
교과서 질문
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.
353
views