Ch. 2 - Limits and Continuity

Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo

Hass 15th Edition

Ch. 2 - Limits and Continuity

Problema 2.6.44

Capitolo 2, Problema 2.6.44

Infinite Limits

Find the limits in Exercises 37–48. Write ∞ or −∞ where appropriate.

lim x→0 (−1) / (x² (x + 1))

Guida verificata passo dopo passo

First, identify the type of limit you are dealing with. Here, you have a limit as x approaches 0 for the function \(-\frac{1}{x^2(x+1)}\).

Examine the behavior of the denominator as x approaches 0. Notice that \(x^2\) approaches 0 and \(x+1\) approaches 1, making the denominator approach 0.

Since the denominator approaches 0, the fraction \(-\frac{1}{x^2(x+1)}\) will tend towards infinity or negative infinity depending on the sign of the numerator and denominator.

Consider the sign of the expression as x approaches 0 from the positive side (x → 0⁺). The denominator \(x^2(x+1)\) is positive, and the numerator is negative, leading to \(-∞\).

Similarly, consider the sign of the expression as x approaches 0 from the negative side (x → 0⁻). The denominator \(x^2(x+1)\) is positive, and the numerator is negative, also leading to \(-∞\).

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:

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

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 points of interest, including points where they may not be defined. In this case, we are interested in the limit as x approaches 0, which requires analyzing the function's values as x gets closer to 0 from both the left and right.

Video consigliato:

05:50

One-Sided Limits

Infinite Limits

Infinite limits occur when the value of a function increases or decreases without bound as the input approaches a certain point. This can result in the limit being expressed as ∞ or −∞. In the given problem, we need to determine if the function approaches positive or negative infinity as x approaches 0, which involves examining the signs and magnitudes of the function's components.

Video consigliato:

05:50

One-Sided Limits

Behavior of Rational Functions

Rational functions are ratios of polynomials, and their limits can often be analyzed by examining the degrees of the polynomials in the numerator and denominator. In this case, the function involves a polynomial in the denominator that approaches zero as x approaches 0, which can lead to infinite limits. Understanding how the numerator and denominator interact is crucial for determining the limit's value.

Video consigliato:

6:04

Intro to Rational Functions

Pratica correlata

Domanda del libro di testo

Use the Intermediate Value Theorem in Exercises 69–74 to prove that each equation has a solution. Then use a graphing calculator or computer grapher to solve the equations.

x³ − 3x − 1 = 0

319

views

rank

Domanda del libro di testo

Use formal definitions to prove the limit statements in Exercises 93–96.

lim x → 0 (−1 / x²) = −∞

302

views

Domanda del libro di testo

Domains and Asymptotes

Determine the domain of each function in Exercises 69–72. Then use various limits to find the asymptotes.