Domanda del libro di testo
At what points are the functions in Exercises 13–30 continuous?
y = (2x – 1)¹/³
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Ch. 2 - Limits and Continuity
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- 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 Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 2, Problema 2.2.53
Suppose limx→c f(x) = 5 and lim x→c g(x) = −2. Find
b. limx→c 2f(x)g(x)
Guida verificata passo dopo passo1
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.
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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.
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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).
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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.
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Infinite Limits
Find the limits in Exercises 37–48. Write ∞ or −∞ where appropriate.
lim x→−5⁻ (3x) / (2x + 10)
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Domanda del libro di testo
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.)
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Domanda del libro di testo
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.
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Domanda del libro di testo
Use formal definitions to prove the limit statements in Exercises 93–96.
lim x → −5 (1 / (x + 5)²) = ∞
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Domanda del libro di testo
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))
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