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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 2.49
Chapter 2, Problem 2.49

Horizontal and Vertical Asymptotes


Determine the domain and range of y = (√16―x²) / (x―2).

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Step 1: Identify the domain of the function. The domain is determined by the values of x for which the function is defined. The function y = (√(16 - x²)) / (x - 2) has two restrictions: the expression under the square root, 16 - x², must be non-negative, and the denominator, x - 2, must not be zero.
Step 2: Solve the inequality 16 - x² ≥ 0 to find the values of x for which the square root is defined. This inequality can be rewritten as x² ≤ 16, which implies -4 ≤ x ≤ 4.
Step 3: Determine the values of x that make the denominator zero. Set x - 2 = 0 and solve for x, which gives x = 2. Since the denominator cannot be zero, x = 2 is excluded from the domain.
Step 4: Combine the results from Steps 2 and 3 to find the domain. The domain is the set of all x such that -4 ≤ x ≤ 4, excluding x = 2. In interval notation, this is [-4, 2) ∪ (2, 4].
Step 5: Determine the range of the function. Consider the behavior of the function as x approaches the endpoints of the domain and the point where the denominator is zero. Analyze the limits as x approaches 2 from the left and right to identify any vertical asymptotes, and consider the maximum and minimum values of the function within the domain to determine the range.

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Key Concepts

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

Domain of a Function

The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. For rational functions, the domain is restricted by values that make the denominator zero. In this case, we need to identify any x-values that would cause the denominator (x - 2) to equal zero, as these values are excluded from the domain.
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Range of a Function

The range of a function is the set of all possible output values (y-values) that the function can produce. To determine the range, we analyze the behavior of the function as x approaches certain critical points, including vertical asymptotes and the limits of the function as x approaches infinity or negative infinity. This helps in understanding the values that y can take.
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Asymptotes

Asymptotes are lines that a graph approaches but never touches. Vertical asymptotes occur where the function is undefined, typically where the denominator is zero. Horizontal asymptotes describe the behavior of the function as x approaches infinity or negative infinity, indicating the value that y approaches. Understanding these concepts is crucial for analyzing the overall behavior of the function.
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Related Practice
Textbook Question

Using the Formal Definition


Prove the limit statements in Exercises 37–50.


lim x→0 x sin (1/x) = 0


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Textbook Question

Domains and Asymptotes


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


y = 4 + 3x² / (x² + 1)

Textbook Question

Using limθ→0 sin θ / θ = 1


Find the limits in Exercises 23–46.


limh→0− h / sin 3h

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Textbook Question

In Exercises 77–80, find a function that satisfies the given conditions and sketch its graph. (The answers here are not unique. Any function that satisfies the conditions is acceptable. Feel free to use formulas defined in pieces if that will help.)


lim x → ±∞ f(x) = 0, lim x → 2⁻ f(x) = ∞, and lim x → 2⁺ f(x) = ∞

Textbook Question

Limits of Rational Functions


In Exercises 13–22, find the limit of each rational function (a) as x → ∞ and (b) as x → −∞. Write ∞ or −∞ where appropriate.


f(x) = (x + 1)/(x² + 3)

Textbook Question

Finding Limits


In Exercises 25–28, find the limit of g(x) as x approaches the indicated value.



lim (4g(x))¹/³ = 2

x →0