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Ch. 2 - Limits and Continuity
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 2 - Limits and ContinuityProblem 56b
Chapter 2, Problem 56b

Find the limits in Exercises 53–58. Write ∞ or −∞ where appropriate.


lim (x² − 1) / (2x + 4) as


b. x→−2⁻

Verified step by step guidance
1
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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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 specific points, including points of discontinuity or infinity. Evaluating limits is essential for determining the continuity of functions and for finding derivatives.
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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.
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One-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.
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Related Practice
Textbook Question

Find the limits in Exercises 53–58. Write ∞ or −∞ where appropriate.


lim (x²/2 − 1/x) as


b. x→0⁻

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Textbook Question

Find the limits in Exercises 53–58. Write ∞ or −∞ where appropriate.


lim (x² − 3x + 2) / (x³ − 4x) as


b. x→−2⁺

190
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Textbook Question

Find the limits in Exercises 53–58. Write ∞ or −∞ where appropriate.


lim (x² − 3x + 2) / (x³ − 2x²) as


a. x→0⁺

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Textbook Question

Find the limits in Exercises 53–58. Write ∞ or −∞ where appropriate.


lim (x²/2 − 1/x) as


d. x→−1

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Textbook Question

[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?

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Textbook Question

Find the limits in Exercises 53–58. Write ∞ or −∞ where appropriate.


lim (x² − 3x + 2) / (x³ − 2x²) as


d. x→2

207
views