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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 2.6.66
Chapter 2, Problem 2.6.66

Graphing Simple Rational Functions


Graph the rational functions in Exercises 63–68. Include the graphs and equations of the asymptotes and dominant terms.


y = −3/(x − 3)

Verified step by step guidance
1
Identify the type of rational function: The given function is y = -3/(x - 3), which is a simple rational function with a single term in the denominator.
Determine the vertical asymptote: Set the denominator equal to zero, x - 3 = 0, which gives x = 3. This is the vertical asymptote of the function.
Determine the horizontal asymptote: Since the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0.
Analyze the behavior near the asymptotes: As x approaches 3 from the left, the function y = -3/(x - 3) tends to negative infinity, and as x approaches 3 from the right, the function tends to positive infinity.
Sketch the graph: Plot the vertical asymptote at x = 3 and the horizontal asymptote at y = 0. Draw the curve approaching these asymptotes, showing the behavior described in the previous step.

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Key Concepts

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

Rational Functions

A rational function is a ratio of two polynomials, typically expressed as f(x) = P(x)/Q(x), where P(x) and Q(x) are polynomials. Understanding the behavior of rational functions involves analyzing their domain, asymptotes, and intercepts. The function y = -3/(x - 3) is a simple rational function with a linear polynomial in the denominator.
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Intro to Rational Functions

Asymptotes

Asymptotes are lines that a graph approaches but never touches. For rational functions, vertical asymptotes occur where the denominator is zero, and horizontal or oblique asymptotes describe end behavior. In y = -3/(x - 3), the vertical asymptote is x = 3, indicating where the function is undefined and the graph approaches infinity.
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Dominant Terms

Dominant terms in a rational function determine its behavior as x approaches infinity or negative infinity. For y = -3/(x - 3), the dominant term is -3/x, which influences the horizontal asymptote. As x becomes very large or very small, the function approaches y = 0, indicating a horizontal asymptote at y = 0.
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Related Practice
Textbook Question

Domains and Asymptotes


Determine the domain of each function in Exercises 69–72. Then use various limits to find the asymptotes.


y = (√(x² + 4)) / x

Textbook Question

Infinite Limits


Find the limits in Exercises 37–48. Write ∞ or −∞ where appropriate.


lim x→0 (−1) / (x² (x + 1))

Textbook Question

Finding Deltas Algebraically


Each of Exercises 15–30 gives a function f(x) and numbers L, c, and ε>0. In each case, find the largest open interval about c on which the inequality |f(x)−L| <ε holds. Then give a value for δ>0 such that for all x satisfying 0 < |x−c| < δ, the inequality |f(x)−L| < ε holds.


f(x) = 1/x, L = 1/4, c = 4, ε = 0.05

Textbook Question

Limits with trigonometric functions


Find the limits in Exercises 43–50.


limx→0 (1 + x + sin x) / (3 cosx)

Textbook Question

Finding Limits of Differences When x → ±∞


Find the limits in Exercises 84–90. (Hint: Try multiplying and dividing by the conjugate.)


lim x → ∞ (√(x² + x) − √(x² − x))

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Textbook Question

Exercises 5–10 refer to the function

f(x) = { x² − 1, −1 ≤ x < 0

2x, 0 < x < 1

1, x = 1

−2x + 4, 1 < x < 2

0, 2 < x < 3

graphed in the accompanying figure.

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At what values of x is f continuous?