Textbook Question
Use the graph of the greatest integer function y = ⌊x⌋, Figure 1.10 in Section 1.1, to help you find the limits in Exercises 21 and 22.
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b. limt→4−(t−⌊t⌋)
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 2.2.78b
Theory and Examples
If limx→−2 f(x) / x² = 1, find
b. limx→−2 f(x) / x
Verified step by step guidance1
First, understand the given limit: \( \lim_{x \to -2} \frac{f(x)}{x^2} = 1 \). This means as \( x \) approaches \( -2 \), the ratio of \( f(x) \) to \( x^2 \) approaches 1.
To find \( \lim_{x \to -2} \frac{f(x)}{x} \), consider the relationship between \( f(x) \) and \( x^2 \). Since \( \lim_{x \to -2} \frac{f(x)}{x^2} = 1 \), we can express \( f(x) \) as \( f(x) = x^2 + g(x) \), where \( g(x) \) is a function that becomes negligible as \( x \) approaches \( -2 \).
Substitute \( f(x) = x^2 + g(x) \) into \( \lim_{x \to -2} \frac{f(x)}{x} \) to get \( \lim_{x \to -2} \frac{x^2 + g(x)}{x} \).
Simplify the expression: \( \lim_{x \to -2} \left( x + \frac{g(x)}{x} \right) \). As \( x \to -2 \), \( \frac{g(x)}{x} \) should approach 0 if \( g(x) \) is negligible compared to \( x^2 \).
Finally, evaluate the limit: \( \lim_{x \to -2} x + \lim_{x \to -2} \frac{g(x)}{x} \). Since \( \lim_{x \to -2} \frac{g(x)}{x} \) approaches 0, the limit simplifies to \( \lim_{x \to -2} x \).
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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. In this case, we are interested in the limit of the function f(x) divided by x² as x approaches -2. Understanding limits is crucial for evaluating the behavior of functions near points of interest, especially when direct substitution may lead to indeterminate forms.
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L'Hôpital's Rule
L'Hôpital's Rule is a method used to evaluate limits of indeterminate forms, such as 0/0 or ∞/∞. It states that if the limit of f(x)/g(x) results in an indeterminate form, the limit can be found by taking the derivative of the numerator and the derivative of the denominator. This rule can simplify the process of finding limits, especially when dealing with rational functions.
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Continuity
Continuity of a function at a point means that the function is defined at that point, the limit exists, and the limit equals the function's value at that point. In the context of the given problem, understanding whether f(x) is continuous at x = -2 will help in determining the limit of f(x)/x as x approaches -2. Continuity ensures that the behavior of the function is predictable near the point of interest.
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Related Practice
Textbook Question
Average Rates of Change
In Exercises 1–6, find the average rate of change of the function over the given interval or intervals.
g(t)=2+cos t
b. [0,π]
Textbook Question
Estimating Limits
[Technology Exercise] You will find a graphing calculator useful for Exercises 67–74.
Let f(x)=(x² − 1)/(|x| − 1).
b. Support your conclusion in part (a) by graphing f near c = -1 and using Zoom and Trace to estimate y-values on the graph as x→−1.
Textbook Question
Estimating Limits
[Technology Exercise] You will find a graphing calculator useful for Exercises 67–74.
Let h(x)=(x² − 2x − 3)/(x² − 4x + 3)
b. Support your conclusions in part (a) by graphing h near c = 3 and using Zoom and Trace to estimate y-values on the graph as x→3.
Textbook Question
Estimating Limits
[Technology Exercise] You will find a graphing calculator useful for Exercises 67–74.
Let G(x)=(x + 6)/(x² + 4x − 12)
b. Support your conclusions in part (a) by graphing G and using Zoom and Trace to estimate y-values on the graph as x→−6.
Textbook Question
Suppose limx→b f(x) = 7 and lim x→b g(x) = −3. Find
b. limx→b f(x)⋅g(x)