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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 2.6.15
Chapter 2, Problem 2.6.15

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)

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Step 1: To find the limit of the rational function \( f(x) = \frac{x + 1}{x^2 + 3} \) as \( x \to \infty \), start by analyzing the degrees of the numerator and the denominator. The numerator \( x + 1 \) is of degree 1, and the denominator \( x^2 + 3 \) is of degree 2.
Step 2: Since the degree of the denominator is greater than the degree of the numerator, the limit as \( x \to \infty \) will be determined by the leading terms. Divide each term in the numerator and the denominator by \( x^2 \), the highest power of \( x \) in the denominator.
Step 3: After dividing, the expression becomes \( \frac{\frac{x}{x^2} + \frac{1}{x^2}}{\frac{x^2}{x^2} + \frac{3}{x^2}} = \frac{\frac{1}{x} + \frac{1}{x^2}}{1 + \frac{3}{x^2}} \). As \( x \to \infty \), both \( \frac{1}{x} \) and \( \frac{1}{x^2} \) approach 0.
Step 4: Therefore, the limit of the function as \( x \to \infty \) is \( \frac{0 + 0}{1 + 0} = 0 \).
Step 5: To find the limit as \( x \to -\infty \), repeat the same process. The behavior of the function is similar because the terms \( \frac{1}{x} \) and \( \frac{1}{x^2} \) also approach 0 as \( x \to -\infty \). Thus, the limit is \( \frac{0 + 0}{1 + 0} = 0 \).

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Key Concepts

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

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. In the context of rational functions, limits help determine the behavior of the function as the variable approaches infinity or negative infinity, which is crucial for understanding end behavior.
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One-Sided Limits

Rational Functions

Rational functions are expressions formed by the ratio of two polynomials. The behavior of these functions can vary significantly based on the degrees of the numerator and denominator, especially as x approaches infinity or negative infinity, which influences the limits we calculate.
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End Behavior

End behavior refers to the behavior of a function as the input values become very large or very small. For rational functions, this is often determined by the leading terms of the numerator and denominator, which dictate whether the limit approaches a finite value, infinity, or negative infinity.
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Related Practice
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→∞ (x − 3) / √(4x² + 25)

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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 → ∞ (√(x + 9) − √(x + 4))

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Textbook Question

Using the Formal Definition


Prove the limit statements in Exercises 37–50.


limx→1 f(x) = 1 if f(x) = {x², x ≠ 1

2, x = 1

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Textbook Question

Use the Intermediate Value Theorem in Exercises 69–74 to prove that each equation has a solution. Then use a graphing calculator or computer grapher to solve the equations.

x³ − 15x + 1 = 0 (three roots)

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Textbook Question

Finding Limits


In Exercises 3–8, find the limit of each function (a) as x → ∞ and (b) as x → −∞. (You may wish to visualize your answer with a graphing calculator or computer.)


f(x) = 2/x − 3

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Textbook Question

Limits of Average Rates of Change


Because of their connection with secant lines, tangents, and instantaneous rates, limits of the form limh→0 (f(x+h) − f(x)) / h occur frequently in calculus. In Exercises 57–62, evaluate this limit for the given value of x and function f.


f(x) = x², x = 1

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