교과서 질문
Find the limits in Exercises 53–58. Write ∞ or −∞ where appropriate.
lim (x² − 3x + 2) / (x³ − 4x) as
b. x→−2⁺
190
views
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 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 59a
Find the limits in Exercises 59–62. Write ∞ or −∞ where appropriate.
lim (2 − 3 / t¹/³) as
a. t → 0⁺
검증된 단계별 안내1
Identify the expression for which you need to find the limit: \( 2 - \frac{3}{t^{1/3}} \).
Recognize that as \( t \to 0^+ \), \( t^{1/3} \to 0^+ \) as well, since the cube root of a positive number approaching zero also approaches zero.
Consider the behavior of the term \( \frac{3}{t^{1/3}} \). As \( t^{1/3} \to 0^+ \), \( \frac{3}{t^{1/3}} \to +\infty \) because dividing by a very small positive number results in a very large positive number.
Analyze the entire expression \( 2 - \frac{3}{t^{1/3}} \). Since \( \frac{3}{t^{1/3}} \to +\infty \), the expression \( 2 - \frac{3}{t^{1/3}} \) will tend towards \(-\infty\).
Conclude that the limit of the expression as \( t \to 0^+ \) is \(-\infty\).
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.
Limits
Limits are fundamental concepts in calculus that describe the behavior of a function as its input approaches a certain value. They help in understanding how functions behave near specific points, including points of discontinuity or infinity. In this case, we are interested in the limit of the function as t approaches 0 from the positive side (0⁺).
추천 영상:
05:50
One-Sided Limits
One-Sided Limits
One-sided limits refer to the value that a function approaches as the input approaches a specific point from one side only, either the left (−) or the right (+). In this question, we are evaluating the right-hand limit as t approaches 0, which is crucial for determining the function's behavior in that region without considering values from the left.
추천 영상:
05:50One-Sided Limits
Cube Root Function
The cube root function, denoted as t¹/³, is a continuous function that returns the number which, when cubed, gives the input value. As t approaches 0, the cube root of t also approaches 0. Understanding how this function behaves near 0 is essential for evaluating the limit in the given problem.
추천 영상:
5:57Graphs of Common Functions
관련 실천
교과서 질문
Find the limits in Exercises 53–58. Write ∞ or −∞ where appropriate.
lim (x² − 3x + 2) / (x³ − 2x²) as
a. x→0⁺
184
views
교과서 질문
Theory and Examples
Suppose that g(x) ≤ f(x) ≤ h(x) for all x≠2 and suppose that lim x→2 g(x) = lim x→2 h(x) = −5. Can we conclude anything about the values of f, g, and h at x = 2? Could f(2) = 0? Could limx→2 f(x)=0? Give reasons for your answers.
310
views
교과서 질문
Theory and Examples
If x⁴ ≤ f(x) ≤ x² for x in [−1,1] and x² ≤ f(x) ≤ x⁴ for x < - 1 and x > 1, at what points c do you automatically know limx→c f(x)? What can you say about the value of the limit at these points?
320
views
교과서 질문
Use the formal definitions from Exercise 97 to prove the limit statements in Exercises 98–102.
lim x→2⁻ (1 / (x − 2)) = −∞
285
views
교과서 질문
Find the limits in Exercises 53–58. Write ∞ or −∞ where appropriate.
lim (x² − 3x + 2) / (x³ − 2x²) as
d. x→2
207
views