교과서 질문
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)
361
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 2.4.43
Using limθ→0 sin θ / θ = 1
Find the limits in Exercises 23–46.
limθ→0 tan θ / θ²cot 3θ
검증된 단계별 안내1
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.
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
2m도움이 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 θ approaches 0 is essential to determine the behavior of the expression tan θ / θ²cot 3θ.
추천 영상:
06:11
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.
추천 영상:
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.
추천 영상:
가이드 코스
5:50Power Rules
관련 실천
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
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.
limx→0 (cos²x − cos x) / x²
262
views
교과서 질문
Limits of quotients
Find the limits in Exercises 23–42.
limx→−5 (x² + 3x − 10) / x + 5
175
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