Textbook Question
Finding One-Sided Limits Algebraically
Find the limits in Exercises 11–20.
limx→1− (1/(x + 1))((x + 6)/x)((3 − x)/7)
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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 2.5.40
Find the limits in Exercises 31–40. Are the functions continuous at the point being approached?
lim ϴ → 0 cos (πϴ/sin ϴ)
Verified step by step guidance1
First, recognize that the limit involves a trigonometric function as ϴ approaches 0. The expression is cos(πϴ/sinϴ).
To evaluate the limit, consider the behavior of the argument of the cosine function, πϴ/sinϴ, as ϴ approaches 0.
Use the trigonometric identity sinϴ ≈ ϴ when ϴ is near 0. This approximation helps simplify the expression πϴ/sinϴ to π, since sinϴ/ϴ approaches 1.
Substitute the simplified argument back into the cosine function: cos(πϴ/sinϴ) becomes cos(π) as ϴ approaches 0.
Evaluate cos(π) to determine the limit. Then, consider the definition of continuity: a function is continuous at a point if the limit as ϴ approaches that point equals the function's value at that point. Check if this condition holds for the given function at ϴ = 0.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Limits
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, including points of discontinuity. In this case, we are interested in the limit of the function as θ approaches 0, which requires evaluating the function's behavior close to that point.
Recommended video:
05:50
One-Sided Limits
Continuity
A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. For the given limit, we need to determine if the function cos(πθ/sin(θ)) is continuous at θ = 0. This involves checking if the limit exists and matches the function's value at that point.
Recommended video:
05:34Intro to Continuity
Trigonometric Limits
Trigonometric limits often involve special techniques or identities to evaluate limits involving sine and cosine functions. In this case, the limit involves the sine function in the denominator, which approaches 0 as θ approaches 0. Understanding how to manipulate trigonometric expressions and apply L'Hôpital's Rule or Taylor series can be crucial for finding the limit accurately.
Recommended video:
6:04Introduction to Trigonometric Functions
Related Practice
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.
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Textbook Question
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)
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Textbook Question
Theory and Examples
If limx→4 (f(x) − 5) / (x − 2) = 1, find limx→4 f(x).
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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 → ∞ (√(x² + 25) − √(x² − 1))
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Textbook Question
Limits as x → ∞ or x → −∞
The process by which we determine limits of rational functions applies equally well to ratios containing noninteger or negative powers of x. Divide numerator and denominator by the highest power of x in the denominator and proceed from there. Find the limits in Exercises 23–36. Write ∞ or −∞ where appropriate.
lim x→∞ (2√x + x⁻¹) / (3x − 7)
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