Skip to main content
Ch. 2 - Limits and Continuity
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 2 - Limits and ContinuityProblem 114b
Chapter 2, Problem 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

Verified step by step guidance
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.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
3m
Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.
Recommended video:
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.
Recommended video:
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.
Recommended video:
06:29
Derivatives Applied To Velocity
Related Practice
Textbook Question

[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

Textbook Question

[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

Textbook Question

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)