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 2.6.44
Infinite Limits
Find the limits in Exercises 37–48. Write ∞ or −∞ where appropriate.
lim x→0 (−1) / (x² (x + 1))
Verified step by step guidance1
First, identify the type of limit you are dealing with. Here, you have a limit as x approaches 0 for the function \(-\frac{1}{x^2(x+1)}\).
Examine the behavior of the denominator as x approaches 0. Notice that \(x^2\) approaches 0 and \(x+1\) approaches 1, making the denominator approach 0.
Since the denominator approaches 0, the fraction \(-\frac{1}{x^2(x+1)}\) will tend towards infinity or negative infinity depending on the sign of the numerator and denominator.
Consider the sign of the expression as x approaches 0 from the positive side (x → 0⁺). The denominator \(x^2(x+1)\) is positive, and the numerator is negative, leading to \(-∞\).
Similarly, consider the sign of the expression as x approaches 0 from the negative side (x → 0⁻). The denominator \(x^2(x+1)\) is positive, and the numerator is negative, also leading to \(-∞\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Limits
A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding how functions behave near points of interest, including points where they may not be defined. In this case, we are interested in the limit as x approaches 0, which requires analyzing the function's values as x gets closer to 0 from both the left and right.
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One-Sided Limits
Infinite Limits
Infinite limits occur when the value of a function increases or decreases without bound as the input approaches a certain point. This can result in the limit being expressed as ∞ or −∞. In the given problem, we need to determine if the function approaches positive or negative infinity as x approaches 0, which involves examining the signs and magnitudes of the function's components.
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05:50One-Sided Limits
Behavior of Rational Functions
Rational functions are ratios of polynomials, and their limits can often be analyzed by examining the degrees of the polynomials in the numerator and denominator. In this case, the function involves a polynomial in the denominator that approaches zero as x approaches 0, which can lead to infinite limits. Understanding how the numerator and denominator interact is crucial for determining the limit's value.
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6:04Intro to Rational Functions
Related Practice
Textbook Question
Domains and Asymptotes
Determine the domain of each function in Exercises 69–72. Then use various limits to find the asymptotes.
y = (√(x² + 4)) / x
Textbook Question
Finding Deltas Algebraically
Each of Exercises 15–30 gives a function f(x) and numbers L, c, and ε>0. In each case, find the largest open interval about c on which the inequality |f(x)−L| <ε holds. Then give a value for δ>0 such that for all x satisfying 0 < |x−c| < δ, the inequality |f(x)−L| < ε holds.
f(x) = 1/x, L = 1/4, c = 4, ε = 0.05
Textbook Question
Graphing Simple Rational Functions
Graph the rational functions in Exercises 63–68. Include the graphs and equations of the asymptotes and dominant terms.
y = −3/(x − 3)
Textbook Question
Limits with trigonometric functions
Find the limits in Exercises 43–50.
limx→0 (1 + x + sin x) / (3 cosx)
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)