Textbook Question
Using the Formal Definition
Prove the limit statements in Exercises 37–50.
limx →4 (9 − x) = 5
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 2.4.45
Using limθ→0 sin θ / θ = 1
Find the limits in Exercises 23–46.
limx→0 (1 − cos 3x) / 2x
Verified step by step guidance1
Recognize that the limit involves a trigonometric function and can be approached using trigonometric identities and the known limit lim(θ→0) (sin θ / θ) = 1.
Rewrite the expression (1 - cos 3x) using the trigonometric identity: 1 - cos A = 2sin²(A/2). In this case, A = 3x, so 1 - cos 3x = 2sin²(3x/2).
Substitute the identity into the limit expression: lim(x→0) (2sin²(3x/2) / 2x). This simplifies to lim(x→0) (sin²(3x/2) / x).
To further simplify, use the substitution u = 3x/2, which implies that as x approaches 0, u also approaches 0. Therefore, x = (2/3)u, and the limit becomes lim(u→0) (sin²(u) / ((2/3)u)).
Apply the limit property lim(u→0) (sin u / u) = 1 to the expression: lim(u→0) (sin²(u) / u) = (lim(u→0) (sin u / u)) * (lim(u→0) sin u) = 1 * 0 = 0. Thus, the original limit evaluates to 0.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Limit Definition
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, particularly where they may not be explicitly defined. The notation lim x→a f(x) indicates the limit of f(x) as x approaches a.
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Trigonometric Limits
Trigonometric limits, such as lim θ→0 (sin θ / θ) = 1, are essential in calculus for evaluating limits involving trigonometric functions. This specific limit is crucial for deriving derivatives of sine and cosine functions and is often used in conjunction with L'Hôpital's Rule or Taylor series expansions to simplify complex limit problems.
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L'Hôpital's Rule
L'Hôpital's Rule is a method for evaluating 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 separately. This rule simplifies the process of finding limits that are otherwise difficult to evaluate directly.
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Related Practice
Textbook Question
Finding One-Sided Limits Algebraically
Find the limits in Exercises 11–20.
limh→0− (√6 − √(5h² + 11h + 6))/ h
Textbook Question
Find the limits in Exercises 31–40. Are the functions continuous at the point being approached?
lim x → π/6 √(csc² x + 5√3 tan x)
Textbook Question
Centering Intervals About a Point
In Exercises 1–6, sketch the interval (a,b), on the x-axis with the point c inside. Then find a value of δ>0 such that a < x < b whenever 0 < |x−c| < δ.
a=4/9, b=4/7, c=1/2
Textbook Question
Using the Formal Definition
Prove the limit statements in Exercises 37–50.
limx→3 (3x − 7) = 2
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 → ∞ (√(9x² − x) − 3x)