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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 54d
Chapter 2, Problem 54d

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


lim x/(x² − 1) as


d. x→−1⁻

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1
Step 1: Understand the problem. We need to find the limit of the function \( \frac{x}{x^2 - 1} \) as \( x \) approaches \( -1 \) from the left (denoted as \( x \to -1^- \)). This involves analyzing the behavior of the function as \( x \) gets very close to \( -1 \) from values less than \( -1 \).
Step 2: Factor the denominator. The expression \( x^2 - 1 \) can be factored using the difference of squares formula: \( x^2 - 1 = (x - 1)(x + 1) \). This will help us understand the behavior of the function near \( x = -1 \).
Step 3: Analyze the sign of the denominator near \( x = -1 \). As \( x \to -1^- \), \( x + 1 \) approaches 0 from the negative side, and \( x - 1 \) is negative. Therefore, the product \( (x - 1)(x + 1) \) is positive.
Step 4: Consider the numerator. The numerator \( x \) approaches \( -1 \) as \( x \to -1^- \). Since \( x \) is negative and the denominator is positive, the overall expression \( \frac{x}{x^2 - 1} \) will be negative.
Step 5: Determine the limit. As \( x \to -1^- \), the denominator \( (x - 1)(x + 1) \) approaches 0, causing the fraction \( \frac{x}{x^2 - 1} \) to become very large in magnitude but negative in sign. Therefore, the limit is \( -\infty \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Limits

Limits are fundamental concepts in calculus that describe the behavior of a function as its input approaches a certain value. They help in understanding how functions behave near points of interest, including points of discontinuity or infinity. In this case, we are interested in the limit of the function as x approaches -1 from the left.
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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 is crucial for analyzing functions that may behave differently from each side of a point, particularly at points of discontinuity or vertical asymptotes.
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One-Sided Limits

Rational Functions

Rational functions are expressions formed by the ratio of two polynomials. The behavior of these functions can vary significantly based on their numerator and denominator. In this limit problem, the function x/(x² - 1) is a rational function, and understanding its structure helps in determining the limit as x approaches -1, especially since the denominator can lead to undefined behavior.
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Intro to Rational Functions
Related Practice
Textbook Question

Find the limits in Exercises 49–52. Write ∞ or −∞ where appropriate.


lim x→(−π/2)⁺ sec x

Textbook Question

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


lim (x²/2 − 1/x) as


b. x→0⁻

Textbook Question

Find the limits in Exercises 49–52. Write ∞ or −∞ where appropriate.


lim θ→0 (2 − cot θ)

Textbook Question

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


lim (x²/2 − 1/x) as


d. x→−1

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?

Textbook Question

[Technology Exercise] In Exercises 33–36, graph the function to see whether it appears to have a continuous extension to the given point a. If it does, use Trace and Zoom to find a good candidate for the extended function’s value at a. If the function does not appear to have a continuous extension, can it be extended to be continuous from the right or left? If so, what do you think the extended function’s value should be?


g(θ) = 5 cos θ / (4θ ― 2π) , a = π/2