교과서 질문
Limits and Continuity
Suppose the functions ƒ(x) and g(x) are defined for all x and that lim (x → 0) ƒ(x) = 1/2 and lim (x → 0) g(x) = √2. Find the limits as x → 0 of the following functions.
e. x + ƒ(x)
269
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Ch. 2 - Limits and Continuity
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- 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 5
Limits and Continuity
In Exercises 5 and 6, find the value that lim (x→0) g(x) must have if the given limit statements hold.
lim ((4―g(x)) / x ) = 1
x→0
검증된 단계별 안내1
Understand the problem: We need to find the value of lim (x→0) g(x) given that lim ((4 - g(x)) / x) = 1 as x approaches 0.
Recall the definition of a limit: The limit of a function as x approaches a certain value is the value that the function approaches as x gets closer to that value.
Set up the equation based on the given limit statement: Since lim ((4 - g(x)) / x) = 1, we can express this as lim (x→0) (4 - g(x)) / x = 1.
Apply the limit property: As x approaches 0, the expression (4 - g(x)) must approach a value such that when divided by x, the result is 1. This implies that (4 - g(x)) must approach x as x approaches 0.
Solve for g(x): Rearrange the equation 4 - g(x) = x to find g(x). As x approaches 0, g(x) must approach 4 to satisfy the limit condition.
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Limits
Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. In this context, we are interested in the limit of the function g(x) as x approaches 0. Understanding limits helps in analyzing the behavior of functions near specific points, which is crucial for solving the given problem.
추천 영상:
05:50
One-Sided Limits
Continuity
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. In this case, continuity is important because it ensures that g(x) behaves predictably as x approaches 0, allowing us to deduce the necessary value of g(0) based on the limit provided.
추천 영상:
05:34Intro to Continuity
L'Hôpital's Rule
L'Hôpital's Rule is a method used to evaluate limits of indeterminate forms, such as 0/0 or ∞/∞. In the given limit expression, if substituting x = 0 leads to an indeterminate form, applying this rule can simplify the limit calculation. This technique is particularly useful when dealing with ratios of functions, as it allows for differentiation to find the limit.
추천 영상:
가이드 코스
5:50Power Rules
관련 실천
교과서 질문
Limits and Continuity
Suppose the functions ƒ(x) and g(x) are defined for all x and that lim (x → 0) ƒ(x) = 1/2 and lim (x → 0) g(x) = √2. Find the limits as x → 0 of the following functions.
f. [ƒ(x) • cos x ] / x―1
255
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교과서 질문
Finding Limits
In Exercises 9–24, find the limit or explain why it does not exist.
lim x→a (x² ― a²)/(x⁴ ― a⁴)
234
views
교과서 질문
Which of the following statements about the function y=f(x) graphed here are true, and which are false?
d. limx→c f(x) exists at every point c in (-1,1).
301
views
교과서 질문
Limits and Continuity
In Exercises 5 and 6, find the value that lim (x→0) g(x) must have if the given limit statements hold.
lim (x lim g(x)) = 2
x→-4 x→0
188
views
교과서 질문
Finding Deltas Graphically
In Exercises 7–14, use the graphs to find a δ>0 such that |f(x)−L| <ε whenever 0< |x−c| <δ.
379
views