Domanda del libro di testo
Find the limits in Exercises 59–62. Write ∞ or −∞ where appropriate.
lim (2 − 3 / t¹/³) as
a. t → 0⁺
221
views
Ch. 2 - Limits and Continuity
Hass15th EditionThomas' CalculusISBN: 9780137616077
Non è quello che usi tu?Cambia libro di testoScegli un libro di testo diverso
TOC.tableOfContents
- 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
Capitolo 2, Problema 58b
Find the limits in Exercises 53–58. Write ∞ or −∞ where appropriate.
lim (x² − 3x + 2) / (x³ − 4x) as
b. x→−2⁺
Guida verificata passo dopo passo1
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.
Risposta video verificata per un problema simile:
Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
4mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
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.
Video consigliato:
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.
Video consigliato:
6:04Intro 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.
Video consigliato:
05:50One-Sided Limits
Pratica correlata
Domanda del libro di testo
Find the limits in Exercises 53–58. Write ∞ or −∞ where appropriate.
lim (x² − 1) / (2x + 4) as
b. x→−2⁻
180
views
Domanda del libro di testo
Find the limits in Exercises 53–58. Write ∞ or −∞ where appropriate.
lim (x² − 3x + 2) / (x³ − 2x²) as
a. x→0⁺
184
views
Domanda del libro di testo
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.
310
views
Domanda del libro di testo
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?
320
views
Domanda del libro di testo
Find the limits in Exercises 53–58. Write ∞ or −∞ where appropriate.
lim (x² − 3x + 2) / (x³ − 2x²) as
d. x→2
207
views