Textbook Question
At what points are the functions in Exercises 13–30 continuous?
y = (2x – 1)¹/³
Ch. 2 - Limits and Continuity
Hass15th EditionThomas' CalculusISBN: 9780137616077
Not the one you use?Change textbookSelect a different textbook
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 2.2.53
Suppose limx→c f(x) = 5 and lim x→c g(x) = −2. Find
b. limx→c 2f(x)g(x)
Verified step by step guidance1
First, understand that the problem involves finding the limit of a product of functions as x approaches a certain value, c.
Recall the limit property for products: if lim(x→c) f(x) = L and lim(x→c) g(x) = M, then lim(x→c) [f(x) * g(x)] = L * M.
In this problem, you are given that lim(x→c) f(x) = 5 and lim(x→c) g(x) = -2.
Apply the limit property for products to find lim(x→c) [2 * f(x) * g(x)]. This can be rewritten as 2 * lim(x→c) [f(x) * g(x)].
Substitute the known limits into the expression: 2 * (5 * -2). Calculate this expression to find the limit.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
1mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Limit of a Function
The limit of a function describes the value that a function approaches as the input approaches a certain point. In this case, limx→c f(x) = 5 indicates that as x gets closer to c, the function f(x) approaches the value 5. Understanding limits is fundamental in calculus as it lays the groundwork for concepts like continuity and derivatives.
Recommended video:
06:11
Limits of Rational Functions: Denominator = 0
Product of Limits
The product of limits states that if the limits of two functions exist as x approaches a certain value, then the limit of their product can be found by multiplying the individual limits. Specifically, if limx→c f(x) = L and limx→c g(x) = M, then limx→c (f(x)g(x)) = L * M. This property is essential for solving the given limit problem involving the functions f(x) and g(x).
Recommended video:
05:18The Product Rule
Constant Multiplication in Limits
When calculating limits, multiplying a function by a constant does not affect the limit's existence. Specifically, if k is a constant and limx→c f(x) = L, then limx→c (k * f(x)) = k * L. This concept is crucial for evaluating the limit limx→c 2f(x)g(x), as the constant 2 can be factored out when calculating the limit.
Recommended video:
05:50One-Sided Limits
Related Practice
Textbook Question
In Exercises 1–4, say whether the function graphed is continuous on [−1, 3]. If not, where does it fail to be continuous and why?
<IMAGE>
1
views
Textbook Question
In Exercises 77–80, find a function that satisfies the given conditions and sketch its graph. (The answers here are not unique. Any function that satisfies the conditions is acceptable. Feel free to use formulas defined in pieces if that will help.)
lim x → ±∞ k(x) = 1, lim x → 1⁻ k(x) = ∞, and lim x → 1⁺ k(x) = −∞
Textbook Question
Using the Sandwich Theorem
a. Suppose that the inequalities 1/2 − x² / 24 < (1 − cos x)/ x² < 1/2 hold for values of x close to zero, except for x = 0 itself. (They do, as you will see in Section 9.9.) What, if anything, does this tell you about limx→0 (1 −cos x)/ x²?
Give reasons for your answer.
[Technology Exercise] b. Graph the equations y=(1/2) − (x²/24), y = (1 - cos x) / x², and y = 1/2 together for −2 ≤ x ≤2. Comment on the behavior of the graphs as x→0.
Textbook Question
Use formal definitions to prove the limit statements in Exercises 93–96.
lim x → −5 (1 / (x + 5)²) = ∞
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.)
g(x) = 1/(2 + (1/x))