교과서 질문
Using the Formal Definition
Prove the limit statements in Exercises 37–50.
lim x→0 x sin (1/x) = 0
<IMAGE>
300
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 2.5.25
At what points are the functions in Exercises 13–30 continuous?
y = √(2x + 3)
검증된 단계별 안내1
Step 1: Understand the concept of continuity. A function is continuous at a point if the limit of the function as it approaches the point from both sides is equal to the function's value at that point.
Step 2: Identify the domain of the function y = √(2x + 3). The expression inside the square root, 2x + 3, must be greater than or equal to zero for the function to be defined.
Step 3: Solve the inequality 2x + 3 ≥ 0 to find the values of x for which the function is defined. This will give you the domain of the function.
Step 4: Since the square root function is continuous wherever it is defined, the function y = √(2x + 3) will be continuous for all x within the domain found in Step 3.
Step 5: Conclude that the function is continuous at all points within the domain determined by the inequality 2x + 3 ≥ 0.
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
1m도움이 되었나요?
주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Continuity of Functions
A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. This means there are no breaks, jumps, or holes in the graph of the function at that point. For a function to be continuous over an interval, it must be continuous at every point within that interval.
추천 영상:
05:34
Intro to Continuity
Domain of a Function
The domain of a function is the set of all possible input values (x-values) for which the function is defined. For the function y = √(2x + 3), the expression under the square root must be non-negative, which imposes restrictions on the values of x. Identifying the domain is crucial for determining where the function is continuous.
추천 영상:
5:10Finding the Domain and Range of a Graph
Square Root Function Properties
The square root function, such as y = √(x), is defined only for non-negative values of x. This means that for the function y = √(2x + 3), we need to ensure that the expression 2x + 3 is greater than or equal to zero. Understanding these properties helps in finding the intervals where the function is continuous.
추천 영상:
06:21Properties of Functions
관련 실천
교과서 질문
Finding Limits
In Exercises 25–28, find the limit of g(x) as x approaches the indicated value.
lim (4g(x))¹/³ = 2
x →0
302
views
교과서 질문
Using limθ→0 sin θ / θ = 1
Find the limits in Exercises 23–46.
limh→0 sin(sin h) / sin h
362
views
교과서 질문
Limits and Infinity
Find the limits in Exercises 37–46.
x⁴ + x³
lim -----------------
x→∞ 12x³ + 128
303
views
교과서 질문
Limits of quotients
Find the limits in Exercises 23–42.
limx→−1 (√(x² + 8) − 3) / (x + 1)
196
views
교과서 질문
At what points are the functions in Exercises 13–30 continuous?
g(x) = { (x² − x – 6)/(x – 3), x ≠ 3
5, x = 3
253
views