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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 107
Capitolo 2, Problema 107

Oblique Asymptotes


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


y = (x² − 1) / x

Guida verificata passo dopo passo
1
Identify the type of asymptotes: Since the degree of the numerator (2) is one more than the degree of the denominator (1), there is an oblique (slant) asymptote.
Find the oblique asymptote: Perform polynomial long division of the numerator \(x^2 - 1\) by the denominator \(x\). The quotient will give the equation of the oblique asymptote.
Perform the division: Divide \(x^2\) by \(x\) to get \(x\). Multiply \(x\) by \(x\) and subtract from \(x^2 - 1\) to get \(-1\). The quotient is \(x\) and the remainder is \(-1\).
Write the equation of the oblique asymptote: The oblique asymptote is given by the equation \(y = x\).
Graph the function: Plot the rational function \(y = \frac{x^2 - 1}{x}\) and the oblique asymptote \(y = x\). Note that the graph will approach the line \(y = x\) as \(x\) approaches infinity or negative infinity.

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Rational Functions

A rational function is a ratio of two polynomials, expressed as f(x) = P(x)/Q(x), where P(x) and Q(x) are polynomials. Understanding the behavior of rational functions is crucial for graphing them, as they can have vertical, horizontal, or oblique asymptotes depending on the degrees of the polynomials involved.
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Intro to Rational Functions

Oblique Asymptotes

Oblique asymptotes occur in rational functions when the degree of the numerator is exactly one higher than the degree of the denominator. They represent a slant line that the graph of the function approaches as x tends to infinity. To find the equation of an oblique asymptote, perform polynomial long division on the rational function.
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Introduction to Cotangent Graph

Polynomial Long Division

Polynomial long division is a method used to divide one polynomial by another, similar to numerical long division. It is essential for finding oblique asymptotes in rational functions, as it helps determine the linear equation that the function approaches. The quotient obtained from the division gives the equation of the oblique asymptote.
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Introduction to Polynomial Functions
Pratica correlata
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 − 4) / (x − 1)

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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 / (x − 1)

217
views
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

Use the formal definitions from Exercise 97 to prove the limit statements in Exercises 98–102.

lim x→2⁻ (1 / (x − 2)) = −∞

285
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Domanda del libro di testo

[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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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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