교과서 질문
Find the limits in Exercises 53–58. Write ∞ or −∞ where appropriate.
lim (x²/2 − 1/x) as
b. x→0⁻
207
views
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 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
2장, 문제 56b
Find the limits in Exercises 53–58. Write ∞ or −∞ where appropriate.
lim (x² − 1) / (2x + 4) as
b. x→−2⁻
검증된 단계별 안내1
Identify the type of limit: This is a one-sided limit as x approaches -2 from the left (x → -2⁻).
Substitute x = -2 into the expression (x² − 1) / (2x + 4) to check if it results in an indeterminate form. Calculate the numerator: (-2)² - 1 = 4 - 1 = 3. Calculate the denominator: 2(-2) + 4 = -4 + 4 = 0. The expression is of the form 3/0, indicating a potential vertical asymptote.
Determine the sign of the expression as x approaches -2 from the left. Choose a test value slightly less than -2, such as x = -2.1. Calculate the denominator: 2(-2.1) + 4 = -4.2 + 4 = -0.2, which is negative.
Since the numerator is positive (3) and the denominator is negative as x approaches -2 from the left, the overall expression approaches negative infinity.
Conclude that the limit of (x² − 1) / (2x + 4) as x approaches -2 from the left is -∞.
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주요 개념
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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 specific points, including points of discontinuity or infinity. Evaluating limits is essential for determining the continuity of functions and for finding derivatives.
추천 영상:
05:50
One-Sided Limits
One-Sided Limits
One-sided limits refer to the value that a function approaches as the input approaches a specific point from one side only, either the left (denoted as x→a⁻) or the right (denoted as x→a⁺). This concept is crucial when dealing with functions that may behave differently from each side of a point, particularly at points of discontinuity.
추천 영상:
05:50One-Sided Limits
Rational Functions
Rational functions are functions that can be expressed as the ratio of two polynomials. Understanding their limits often involves analyzing the behavior of the numerator and denominator as the variable approaches a certain value. In this case, identifying how the function behaves as x approaches -2 is key to determining the limit.
추천 영상:
6:04Intro to Rational Functions
관련 실천
교과서 질문
Find the limits in Exercises 53–58. Write ∞ or −∞ where appropriate.
lim (x² − 3x + 2) / (x³ − 4x) as
b. x→−2⁺
190
views
교과서 질문
Find the limits in Exercises 53–58. Write ∞ or −∞ where appropriate.
lim (x² − 3x + 2) / (x³ − 2x²) as
a. x→0⁺
184
views
교과서 질문
Find the limits in Exercises 53–58. Write ∞ or −∞ where appropriate.
lim (x²/2 − 1/x) as
d. x→−1
161
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교과서 질문
[Technology Exercise] Grinding engine cylinders Before contracting to grind engine cylinders to a cross-sectional area of 9in², you need to know how much deviation from the ideal cylinder diameter of c = 3.385in. you can allow and still have the area come within 0.01in² of the required 9in². To find out, you let A=π(x/2)² and look for the largest interval in which you must hold x to make |A − 9| ≤ 0.01. What interval do you find?
306
views
교과서 질문
Find the limits in Exercises 53–58. Write ∞ or −∞ where appropriate.
lim (x² − 3x + 2) / (x³ − 2x²) as
d. x→2
207
views