Textbook Question
Finding Limits
In Exercises 25–28, find the limit of g(x) as x approaches the indicated value.
lim (4g(x))¹/³ = 2
x →0
302
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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 2.4.35
Using limθ→0 sin θ / θ = 1
Find the limits in Exercises 23–46.
limt→0 sin(1 − cos t) / (1 − cos t)
Verified step by step guidance1
Recognize that the limit involves the expression sin(1 - cos t) / (1 - cos t) as t approaches 0. This is a form that can be related to the standard limit lim(θ→0) sin(θ)/θ = 1.
To use the standard limit, we need to manipulate the expression to match the form sin(θ)/θ. Notice that both the numerator and the denominator have the same expression (1 - cos t).
Apply the standard limit by setting θ = 1 - cos t. As t approaches 0, cos t approaches 1, making θ approach 0.
Rewrite the limit in terms of θ: lim(t→0) sin(1 - cos t) / (1 - cos t) becomes lim(θ→0) sin(θ)/θ.
Since lim(θ→0) sin(θ)/θ = 1, conclude that the original limit also equals 1.
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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 value that f(x) approaches as x gets closer to a.
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Definition of the Definite Integral
Trigonometric Limits
Trigonometric limits, such as lim θ→0 sin θ / θ = 1, are specific cases that often arise in calculus. This particular limit is crucial for evaluating limits involving sine and cosine functions, especially when they appear in indeterminate forms. Understanding this limit allows for simplification of expressions involving trigonometric functions as they approach zero.
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L'Hôpital's Rule
L'Hôpital's Rule is a method used to evaluate limits of indeterminate forms like 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 is particularly useful when dealing with limits involving trigonometric functions and can simplify complex limit problems.
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Related Practice
Textbook Question
At what points are the functions in Exercises 13–30 continuous?
y = √(2x + 3)
226
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Textbook Question
Limits and Infinity
Find the limits in Exercises 37–46.
x⁴ + x³
lim -----------------
x→∞ 12x³ + 128
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Textbook Question
At what points are the functions in Exercises 13–30 continuous?
f(x) = { (x³ − 8)/(x² − 4), x ≠ 2, x ≠ −2
3, x = 2
4, x = −2
209
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Textbook Question
Limits of Average Rates of Change
Because of their connection with secant lines, tangents, and instantaneous rates, limits of the form limh→0 (f(x+h) − f(x)) / h occur frequently in calculus. In Exercises 57–62, evaluate this limit for the given value of x and function f.
f(x) = x², x = 1
333
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Textbook Question
Limits with trigonometric functions
Find the limits in Exercises 43–50.
limx→−π √(x + 4) cos(x + π)
262
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