교과서 질문
Determine the interval(s) on which the following functions are continuous.
f(x)=x^5+6x+17 / x^2−9
976
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Ch. 2 - Limits
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
2장, 문제 2.7.59
Use the definitions given in Exercise 57 to prove the following infinite limits.
lim x→1^- 1 / 1 − x=∞
검증된 단계별 안내1
Understand the problem: We need to prove that the limit of the function \( \frac{1}{1-x} \) as \( x \) approaches 1 from the left (denoted as \( x \to 1^- \)) is infinity.
Consider the behavior of the function \( \frac{1}{1-x} \) as \( x \) approaches 1 from the left. As \( x \) gets closer to 1 from values less than 1, the denominator \( 1-x \) becomes a very small positive number.
Recognize that as \( 1-x \) approaches zero from the positive side, the fraction \( \frac{1}{1-x} \) becomes very large, since dividing by a smaller and smaller positive number results in a larger and larger value.
Formally, for any large positive number \( M \), we need to find a \( \delta > 0 \) such that if \( 0 < 1-x < \delta \), then \( \frac{1}{1-x} > M \). This is the definition of the limit approaching infinity.
Choose \( \delta = \frac{1}{M} \). Then, if \( 0 < 1-x < \delta \), it follows that \( \frac{1}{1-x} > M \), thus proving that \( \lim_{x \to 1^-} \frac{1}{1-x} = \infty \).
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Infinite Limits
Infinite limits describe the behavior of a function as it approaches a certain point, where the function's value increases or decreases without bound. In this case, as x approaches 1 from the left, the function 1 / (1 - x) tends to infinity, indicating that the values of the function grow larger and larger.
추천 영상:
05:50
One-Sided Limits
One-Sided Limits
One-sided limits evaluate the behavior of a function as it approaches a specific point from one direction only. The notation lim x→1^- indicates that we are considering values of x that are less than 1, which is crucial for understanding how the function behaves as it nears the point of interest.
추천 영상:
05:50One-Sided Limits
Continuity and Discontinuity
Continuity refers to a function being unbroken and having no gaps at a point. In this case, the function 1 / (1 - x) is discontinuous at x = 1, as it approaches infinity from the left. Understanding this concept helps clarify why the limit diverges rather than converges to a finite value.
추천 영상:
05:34Intro to Continuity
관련 실천
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