Textbook Question
Limits and Infinity
Find the limits in Exercises 37–46.
2x² + 3
lim -------------
x→⁻∞ 5x² + 7
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 2.5.72
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)
Verified step by step guidance1
Understand the Intermediate Value Theorem (IVT): It states that if a continuous function f(x) takes on values of opposite signs at two points a and b, then there exists at least one c in the interval (a, b) such that f(c) = 0.
Identify the function: Here, the function is f(x) = x³ − 15x + 1. This is a polynomial function, which is continuous everywhere.
Choose appropriate intervals: To apply the IVT, we need to find intervals where the function changes sign. Start by evaluating f(x) at various points to find intervals where f(x) changes from positive to negative or vice versa.
Evaluate the function at chosen points: Calculate f(x) at several points, such as f(-3), f(0), f(3), etc., to find intervals where the sign changes. For example, if f(-3) < 0 and f(0) > 0, then there is a root in the interval (-3, 0). Repeat this process to find all intervals where the function changes sign.
Use a graphing calculator or computer grapher: Once you have identified the intervals using the IVT, use a graphing tool to visualize the function and find the approximate values of the roots within those intervals.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Intermediate Value Theorem
The Intermediate Value Theorem states that if a function is continuous on a closed interval [a, b] and takes on different signs at the endpoints (f(a) and f(b)), then there exists at least one c in (a, b) such that f(c) = 0. This theorem is fundamental for proving the existence of roots in continuous functions.
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06:11
Fundamental Theorem of Calculus Part 1
Continuous Functions
A continuous function is one where small changes in the input result in small changes in the output. Formally, a function f is continuous at a point x = c if the limit of f as x approaches c equals f(c). Understanding continuity is crucial for applying the Intermediate Value Theorem effectively.
Recommended video:
05:34Intro to Continuity
Graphing Techniques
Graphing techniques involve using tools like graphing calculators or software to visualize functions. By plotting the function, one can identify where it crosses the x-axis, indicating the roots of the equation. This visual approach complements analytical methods like the Intermediate Value Theorem in finding solutions.
Recommended video:
06:15Graphing The Derivative
Related Practice
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))
Textbook Question
Using the Sandwich Theorem
a. It can be shown that the inequalities 1 − x²/ 6 < (x sin x) / (2−2cos x) < 1 hold for all values of x close to zero (except for x = 0). What, if anything, does this tell you about limx→0 (x sin x) / (2 − 2cos x)?
Give reasons for your answer.
[Technology Exercise] b. Graph y = 1 − (x²/6), y=(x sinx)/(2 − 2cos x), and y = 1 together for −2 ≤ x ≤2. Comment on the behavior of the graphs as x→0.
Textbook Question
Domains and Asymptotes
Determine the domain of each function in Exercises 69–72. Then use various limits to find the asymptotes.
y = 2x / (x² − 1)
Textbook Question
If functions f(x) and g(x) are continuous for 0 ≤ x ≤ 1, could f(x)/g(x) possibly be discontinuous at a point of [0,1]? Give reasons for your answer.
Textbook Question
Using the Formal Definition
Prove the limit statements in Exercises 37–50.
lim x→0 x² sin (1/x) = 0
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