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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 2.50
Chapter 2, Problem 2.50

Horizontal and Vertical Asymptotes
Assume that constants a and b are positive. Find equations for all horizontal and vertical asymptotes for the graph of y = (√ax² + 4) / (x―b) .

Verified step by step guidance
1
To find the vertical asymptotes, set the denominator equal to zero and solve for x. This is because vertical asymptotes occur where the function is undefined. So, solve the equation x - b = 0.
The solution to x - b = 0 is x = b. Therefore, there is a vertical asymptote at x = b.
To find the horizontal asymptotes, analyze the behavior of the function as x approaches infinity or negative infinity. Consider the leading terms in the numerator and denominator.
The leading term in the numerator is √(ax²), which simplifies to √a * x. The leading term in the denominator is x. As x approaches infinity, the function behaves like (√a * x) / x.
Simplify (√a * x) / x to √a. Therefore, the horizontal asymptote is y = √a.

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Key Concepts

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

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a function as the input approaches infinity or negative infinity. For rational functions, they can often be determined by comparing the degrees of the numerator and denominator. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0. If they are equal, the asymptote is the ratio of the leading coefficients.
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Vertical Asymptotes

Vertical asymptotes occur where a function approaches infinity, typically at values of x that make the denominator zero while the numerator is non-zero. To find vertical asymptotes, set the denominator equal to zero and solve for x. These points indicate where the function is undefined and can lead to infinite behavior in the graph.
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Rational Functions

A rational function is a function represented by the ratio of two polynomials. The behavior of rational functions, particularly their asymptotic behavior, is influenced by the degrees and coefficients of the polynomials in the numerator and denominator. Understanding the structure of rational functions is essential for analyzing their graphs, including identifying asymptotes.
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Related Practice
Textbook Question

In Exercises 1–4, say whether the function graphed is continuous on [−1, 3]. If not, where does it fail to be continuous and why?

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Textbook Question

Using the Formal Definition


Each of Exercises 31–36 gives a function f(x), a point c, and a positive number ε. Find L = lim x→c f(x). Then find a number δ > 0 such that |f(x)−L| < ε whenever 0 < |x−c| < δ.


f(x) = −3x − 2, c = −1, ε = 0.03

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 → −∞ (2x + √(4x² + 3x − 2))

Textbook Question

Slope of a Curve at a Point


In Exercises 7–18, use the method in Example 3 to find (a) the slope of the curve at the given point P, and (b) an equation of the tangent line at P.


y=7−x², P(2,3)

Textbook Question

Using the Sandwich Theorem


If 2−x² ≤ g(x) ≤ 2cosx for all x, find limx→0 g(x).

Textbook Question

Limits as x → ∞ or x → −∞


The process by which we determine limits of rational functions applies equally well to ratios containing noninteger or negative powers of x. Divide numerator and denominator by the highest power of x in the denominator and proceed from there. Find the limits in Exercises 23–36. Write ∞ or −∞ where appropriate.


lim x→∞ √(x² + 1) / (x + 1)