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Ch. 2 - Limits and Continuity
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Hass 15th EditionCh. 2 - Limits and Continuity문제 114b
2장, 문제 114b

[Technology Exercise] Graph the functions in Exercises 113 and 114. Then answer the following questions.


b. How does the graph behave as x → ±∞?


Give reasons for your answers.


y = (3/2)(x / (x − 1))²/³

검증된 단계별 안내
1
Step 1: Begin by understanding the function y = \( \frac{3}{2} \left( \frac{x}{x - 1} \right)^{\frac{2}{3}} \). This is a rational function raised to a fractional power, which affects its graph and behavior.
Step 2: Identify the domain of the function. The expression \( \frac{x}{x - 1} \) is undefined when x = 1, so the domain excludes x = 1. Consider the behavior of the function as x approaches 1 from both sides.
Step 3: Analyze the behavior of the function as x approaches ±∞. As x → ∞, the term \( \frac{x}{x - 1} \) approaches 1, and thus \( \left( \frac{x}{x - 1} \right)^{\frac{2}{3}} \) approaches 1. Similarly, as x → -∞, the term \( \frac{x}{x - 1} \) approaches 1, and the function behaves similarly.
Step 4: Consider the vertical asymptote at x = 1. As x approaches 1 from the left, \( \frac{x}{x - 1} \) becomes very large negatively, and as x approaches 1 from the right, \( \frac{x}{x - 1} \) becomes very large positively. This affects the graph's behavior near x = 1.
Step 5: Graph the function using technology to visualize its behavior. Observe the horizontal asymptote as x → ±∞ and the vertical asymptote at x = 1. The graph should show how the function approaches these asymptotes.

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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.

End Behavior of Functions

The end behavior of a function describes how the function behaves as the input values (x) approach positive or negative infinity. For rational functions, this often involves analyzing the degrees of the numerator and denominator to determine horizontal or oblique asymptotes, which indicate the function's behavior at extreme values of x.
추천 영상:
5:46
Graphs of Exponential Functions

Rational Functions

A rational function is a ratio of two polynomials. The behavior of these functions is influenced by the degrees of the polynomials in the numerator and denominator. Key features include vertical asymptotes, which occur where the denominator is zero, and horizontal or oblique asymptotes, which describe the function's end behavior.
추천 영상:
6:04
Intro to Rational Functions

Asymptotic Analysis

Asymptotic analysis involves studying the behavior of functions as they approach certain limits, such as infinity. For the given function, identifying vertical and horizontal asymptotes helps understand how the function behaves near these limits, providing insight into its long-term behavior as x approaches positive or negative infinity.
추천 영상:
06:29
Derivatives Applied To Velocity
관련 실천
교과서 질문

[Technology Exercise] Graph the functions in Exercises 113 and 114. Then answer the following questions.


a. How does the graph behave as x → 0⁺?


Give reasons for your answers.


y = (3/2)(x − (1 / x))²/³

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교과서 질문

[Technology Exercise] Graph the functions in Exercises 113 and 114. Then answer the following questions.


c. How does the graph behave near x = 1 and x = −1?


Give reasons for your answers.


y = (3/2)(x − (1 / x))²/³

236
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교과서 질문

Additional Graphing Exercises


[Technology Exercise] Graph the curves in Exercises 109–112. Explain the relationship between the curve’s formula and what you see.


y = −1 / √(4 − x²)

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