교과서 질문
Using the Formal Definition
Prove the limit statements in Exercises 37–50.
limx →4 (9 − x) = 5
278
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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
2장, 문제 2.2.49
Limits with trigonometric functions
Find the limits in Exercises 43–50.
limx→−π √(x + 4) cos(x + π)
검증된 단계별 안내1
First, understand the problem: We need to find the limit of the function \( \sqrt{x + 4} \cos(x + \pi) \) as \( x \) approaches \( -\pi \).
Substitute \( x = -\pi \) into the expression \( \sqrt{x + 4} \). This gives \( \sqrt{-\pi + 4} \). Calculate \( -\pi + 4 \) to find the value inside the square root.
Next, substitute \( x = -\pi \) into the expression \( \cos(x + \pi) \). This simplifies to \( \cos(-\pi + \pi) = \cos(0) \). Recall that \( \cos(0) = 1 \).
Combine the results from the previous steps: The limit expression becomes \( \sqrt{-\pi + 4} \times 1 \).
Evaluate \( \sqrt{-\pi + 4} \) to find the final limit value. Ensure that the expression inside the square root is non-negative, as the square root function is defined for non-negative values.
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Limits
Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. They are essential for understanding continuity, derivatives, and integrals. In this context, evaluating the limit as x approaches -π involves determining the behavior of the function near that point.
추천 영상:
05:50
One-Sided Limits
Trigonometric Functions
Trigonometric functions, such as cosine, are periodic functions that relate angles to ratios of sides in right triangles. They play a crucial role in calculus, especially when dealing with limits, derivatives, and integrals involving angles. Understanding how these functions behave near specific points is key to solving limit problems.
추천 영상:
6:04Introduction to Trigonometric Functions
Square Root Function
The square root function, denoted as √(x), is defined for non-negative values of x and represents the principal square root. In limit problems, it is important to consider the domain of the function and how it behaves as the input approaches certain values. In this case, √(x + 4) must be evaluated as x approaches -π to ensure the expression remains valid.
추천 영상:
7:24Multiplying & Dividing Functions
관련 실천
교과서 질문
Domains and Asymptotes
Determine the domain of each function in Exercises 69–72. Then use various limits to find the asymptotes.
y = 4 + 3x² / (x² + 1)
336
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교과서 질문
Finding One-Sided Limits Algebraically
Find the limits in Exercises 11–20.
limh→0− (√6 − √(5h² + 11h + 6))/ h
265
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교과서 질문
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→∞ (x − 3) / √(4x² + 25)
348
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교과서 질문
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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교과서 질문
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)
210
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