Skip to main content
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 58b
Chapter 2, Problem 58b

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


lim (x² − 3x + 2) / (x³ − 4x) as


b. x→−2⁺

Verified step by step guidance
1
Step 1: Identify the type of limit problem. This is a limit problem where we need to find the limit of a rational function as x approaches -2 from the right (x → -2⁺).
Step 2: Analyze the behavior of the numerator and denominator separately as x approaches -2 from the right. The numerator is x² - 3x + 2, and the denominator is x³ - 4x.
Step 3: Substitute x = -2 into the numerator and denominator to check if they approach zero or any other value. For the numerator, calculate (-2)² - 3(-2) + 2. For the denominator, calculate (-2)³ - 4(-2).
Step 4: Determine if the limit results in an indeterminate form like 0/0 or ∞/∞. If it does, consider using algebraic manipulation or L'Hôpital's Rule to resolve the indeterminate form.
Step 5: If the limit is not indeterminate, evaluate the behavior of the function as x approaches -2 from the right. Consider the signs of the numerator and denominator to determine if the limit approaches ∞, -∞, or a finite value.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
4m
Was this helpful?

Key Concepts

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

Limits

A limit in calculus describes the value that a function approaches as the input approaches a certain point. Understanding limits is crucial for analyzing the behavior of functions at specific points, especially where they may not be explicitly defined. In this problem, we are interested in the behavior of the function as x approaches -2 from the right.
Recommended video:
05:50
One-Sided Limits

Rational Functions

Rational functions are ratios of two polynomials. The behavior of these functions near certain points, such as where the denominator is zero, can lead to undefined values or vertical asymptotes. Analyzing the limit of a rational function often involves simplifying the expression or using algebraic techniques to understand its behavior near critical points.
Recommended video:
6:04
Intro to Rational Functions

One-Sided Limits

One-sided limits consider the behavior of a function as the input approaches a specific value from one side—either from the left (x→a⁻) or the right (x→a⁺). In this problem, x→−2⁺ indicates that we are examining the limit as x approaches -2 from values greater than -2, which can reveal different behavior compared to approaching from the left.
Recommended video:
05:50
One-Sided Limits
Related Practice
Textbook Question

Find the limits in Exercises 59–62. Write ∞ or −∞ where appropriate.


lim (2 − 3 / t¹/³) as


a. t → 0⁺

Textbook Question

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


lim (x² − 1) / (2x + 4) as


b. x→−2⁻

Textbook Question

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


lim (x² − 3x + 2) / (x³ − 2x²) as


a. x→0⁺

Textbook Question

Theory and Examples


Suppose that g(x) ≤ f(x) ≤ h(x) for all x≠2 and suppose that lim x→2 g(x) = lim x→2 h(x) = −5. Can we conclude anything about the values of f, g, and h at x = 2? Could f(2) = 0? Could limx→2 f(x)=0? Give reasons for your answers.

Textbook Question

Theory and Examples


If x⁴ ≤ f(x) ≤ x² for x in [−1,1] and x² ≤ f(x) ≤ x⁴ for x < - 1 and x > 1, at what points c do you automatically know limx→c f(x)? What can you say about the value of the limit at these points?

Textbook Question

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


lim (x² − 3x + 2) / (x³ − 2x²) as


d. x→2