Textbook Question
Theory and Examples
Suppose that f is an odd function of x. Does knowing that limx→0+ f(x) = 3 tell you anything about limx→0− f(x)? Give reasons for your answer.
Ch. 2 - Limits and Continuity
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 2, Problem 2.6.13
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) = (2x + 3)/(5x + 7)
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Identify the degrees of the numerator and the denominator. In the function f(x) = (2x + 3)/(5x + 7), both the numerator and the denominator are linear polynomials with degree 1.
For rational functions where the degrees of the numerator and denominator are equal, the limit as x approaches infinity or negative infinity is the ratio of the leading coefficients.
The leading coefficient of the numerator is 2, and the leading coefficient of the denominator is 5.
Calculate the limit as x approaches infinity: lim(x→∞) f(x) = 2/5.
Calculate the limit as x approaches negative infinity: lim(x→−∞) f(x) = 2/5.
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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 as x approaches infinity or negative infinity can often be simplified by comparing the degrees of the numerator and denominator, which influences the limit's value.
Recommended video:
6:04Intro to Rational Functions
End Behavior
End behavior refers to the behavior of a function as the input values become very large or very small. For rational functions, analyzing the leading coefficients and degrees of the numerator and denominator allows us to predict whether the limit will approach a finite number, infinity, or negative infinity.
Recommended video:
03:07Cases Where Limits Do Not Exist
Related Practice
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=x³−3x²+4, P(2,0)
Textbook Question
Limits of quotients
Find the limits in Exercises 23–42.
limx→−5 (x² + 3x − 10) / x + 5
Textbook Question
Explain why the equation cos x = x has at least one solution.
Textbook Question
Suppose that a function f(x) is defined for all real values of x except x=c. Can anything be said about the existence of limx→c f(x)? Give reasons for your answer.
Textbook Question
Removable discontinuity Give an example of a function f (x) that is continuous for all values of x except x = 2, where it has a removable discontinuity. Explain how you know that f is discontinuous at x = 2, and how you know the discontinuity is removable.