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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 2.47c
Chapter 2, Problem 2.47c

Horizontal and Vertical Asymptotes


Use limits to determine the equations for all vertical asymptotes.


x² + x ― 6
c. y = ------------------
x² + 2x ― 8

Verified step by step guidance
1
Step 1: Identify the vertical asymptotes by setting the denominator equal to zero. Solve the equation \(x^2 + 2x - 8 = 0\) to find the values of \(x\) where the function is undefined.
Step 2: Factor the quadratic equation \(x^2 + 2x - 8\) to find its roots. This can be done by finding two numbers that multiply to -8 and add to 2.
Step 3: Once the roots are found, these values of \(x\) are the vertical asymptotes. Use limits to confirm that the function approaches infinity as \(x\) approaches these values from either side.
Step 4: To find horizontal asymptotes, compare the degrees of the numerator and the denominator. Since both are quadratic, divide the leading coefficients of \(x^2\) terms.
Step 5: Use limits to verify the horizontal asymptote by evaluating \(\lim_{x \to \infty} \frac{x^2 + x - 6}{x^2 + 2x - 8}\) and \(\lim_{x \to -\infty} \frac{x^2 + x - 6}{x^2 + 2x - 8}\).

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Key Concepts

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

Vertical Asymptotes

Vertical asymptotes occur in a rational function when the denominator approaches zero while the numerator does not. This indicates that the function's value increases or decreases without bound as it approaches a specific x-value. To find vertical asymptotes, we set the denominator equal to zero and solve for x.
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Introduction to Cotangent Graph Example 1

Limits

Limits are a fundamental concept in calculus that describe the behavior of a function as it approaches a particular point. They are essential for analyzing the values of functions near points of discontinuity, such as vertical asymptotes. By evaluating limits, we can determine the function's behavior as it approaches the asymptote from either side.
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Rational Functions

Rational functions are expressions formed by the ratio of two polynomials. They can exhibit unique behaviors, such as vertical and horizontal asymptotes, depending on the degrees of the numerator and denominator. Understanding the structure of rational functions is crucial for identifying asymptotes and analyzing their limits.
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Related Practice
Textbook Question

Limits and Continuity


On what intervals are the following functions continuous?


b. g(x) = csc x

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Textbook Question

Infinite Limits


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


b. lim x→0⁻ 2 / (3x¹/³)

Textbook Question

Limits and Continuity


On what intervals are the following functions continuous?


c. h(x) = cos x / x―π

Textbook Question

Horizontal and Vertical Asymptotes


Use limits to determine the equations for all horizontal asymptotes.

_____

√x² + 4

c. g(x) = -----------

x

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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b. Does lim x → −1⁺ f (x) exist?

Textbook Question

Suppose limx→4 f(x) = 0 and lim x→4 g(x) = −3. Find


c. limx→4 (g(x))²