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Ch. 2 - Limits and Continuity
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 2 - Limits and ContinuityProblema 113c
Capitolo 2, Problema 113c

[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

Guida verificata passo dopo passo
1
Step 1: Begin by understanding the function y = (3/2)(x - (1/x))^(2/3). This is a composite function involving a rational expression inside a fractional power. The expression inside the power is (x - (1/x)), which can be simplified further to understand its behavior.
Step 2: Analyze the behavior of the function as x approaches 1 and -1. For x = 1, substitute x = 1 into the expression (x - (1/x)) to get (1 - 1) = 0. Therefore, the function y = (3/2)(0)^(2/3) = 0. This indicates that the graph passes through the point (1, 0).
Step 3: For x = -1, substitute x = -1 into the expression (x - (1/x)) to get (-1 - (-1)) = 0. Similarly, the function y = (3/2)(0)^(2/3) = 0. This indicates that the graph also passes through the point (-1, 0).
Step 4: Consider the derivative of the function to understand the behavior near x = 1 and x = -1. The derivative will help determine if there are any local maxima, minima, or points of inflection near these values. Calculate the derivative using the chain rule and power rule.
Step 5: Examine the limits of the function as x approaches 1 and -1 from both sides. This will help determine if there are any asymptotic behaviors or discontinuities. Consider the limit of (x - (1/x)) as x approaches 1 and -1, and how the fractional power affects the overall function.

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Graphing Functions

Graphing functions involves plotting points on a coordinate plane to visualize the behavior of a function. This helps in understanding how the function behaves at different values of x, including identifying key features such as intercepts, asymptotes, and critical points. Technology tools like graphing calculators or software can assist in accurately plotting complex functions.
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Graph of Sine and Cosine Function

Behavior Near Specific Points

Analyzing the behavior of a graph near specific points, such as x = 1 and x = -1, involves examining the function's limits and continuity. This includes determining if the function approaches a particular value, becomes undefined, or exhibits asymptotic behavior. Understanding these behaviors is crucial for predicting the function's behavior in the vicinity of these points.
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Critical Points

Fractional Exponents

Fractional exponents, such as ²/³, indicate roots and powers simultaneously. In the function y = (3/2)(x − (1 / x))²/³, the exponent affects the shape and smoothness of the graph. It implies taking the cube root and then squaring the result, which can lead to unique graph characteristics, especially near points where the base expression approaches zero or infinity.
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Introduction to Exponent Rules
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Domanda del libro di testo

[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

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Domanda del libro di testo

Oblique Asymptotes


Graph the rational functions in Exercises 103–108. Include the graphs and equations of the asymptotes.


y = (x² − 1) / x

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Domanda del libro di testo

[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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Domanda del libro di testo

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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