Skip to main content
Ch. 2 - Limits
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 2 - LimitsProblema 14
Capitolo 2, Problema 14

Determine the following limits.
lim x→a (3x + 1)^2 − (3a + 1)^2 / x − a, where a is constant

Guida verificata passo dopo passo
1
Recognize that the given limit is in the indeterminate form 0/0 when x approaches a, which suggests using algebraic manipulation or L'Hôpital's Rule.
Notice that the expression (3x + 1)^2 − (3a + 1)^2 is a difference of squares, which can be factored as [(3x + 1) - (3a + 1)][(3x + 1) + (3a + 1)].
Simplify the difference of squares: [(3x + 1) - (3a + 1)] becomes 3(x - a) and [(3x + 1) + (3a + 1)] becomes 6x + 6a + 2.
Substitute the factored form into the limit expression: lim x→a [3(x - a)(6x + 6a + 2)] / (x - a).
Cancel the (x - a) terms in the numerator and denominator, then evaluate the limit of the remaining expression as x approaches a.

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:
4m

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. In this case, we are interested in the limit of the expression as x approaches a. Understanding limits is crucial for analyzing continuity, derivatives, and integrals, as they form the foundation for these concepts.
Video consigliato:
05:50
One-Sided Limits

Difference of Squares

The difference of squares is an algebraic identity that states a^2 - b^2 = (a - b)(a + b). This identity can simplify expressions involving squares, making it easier to evaluate limits. In the given limit problem, recognizing that the expression can be factored using this identity will help in simplifying the limit calculation.
Video consigliato:
Percorso guidato
05:13
Slopes of Tangent Lines

L'Hôpital's Rule

L'Hôpital's Rule is a method for evaluating limits that result in indeterminate forms, such as 0/0 or ∞/∞. It states that if the limit of f(x)/g(x) leads to an indeterminate form, the limit can be found by taking the derivative of the numerator and the derivative of the denominator. This rule is particularly useful in the context of the given limit problem, where direct substitution leads to an indeterminate form.
Video consigliato:
5:50
Power Rules
Pratica correlata
Domanda del libro di testo

The position of an object moving vertically along a line is given by the function s(t)=−4.9t2+30t+20s\(\left\)(t\(\right\))=-4.9t^2+30t+20. Find the average velocity of the object over the following intervals.

[0,3]\(\left\[\lbrack\)0,3\(\right\]\rbrack\)

359
views
Domanda del libro di testo

The position of an object moving vertically along a line is given by the function s(t)=−16t2+128ts\(\left\)(t\(\right\))=-16t^2+128t. Find the average velocity of the object over the following intervals.

[1,4]\(\left\[\lbrack\)1,4\(\right\]\rbrack\)

378
views
Domanda del libro di testo

Determine the following limits.

lim h→0 (h + 6)^2 + (h + 6) − 42 / h

358
views
Domanda del libro di testo

Let g(x)=x3−4x8∣x−2∣g\(\left\)(x\(\right\))=\(\frac{x^3-4x}{8\left|x-2\right|}\). <IMAGE>

Make a conjecture about the values of limx→2−g(x){\(\displaystyle\)\(\lim\)_{x\(\to\)2^{-}}g\(\left\)(x\(\right\))}, limx→2+g(x){\(\displaystyle\)\(\lim\)_{x\(\to\)2^{+}}g\(\left\)(x\(\right\))}, and limx→2g(x){\(\displaystyle\)\(\lim\)_{x\(\to\)2}g\(\left\)(x\(\right\))} or state that they do not exist.

358
views
Domanda del libro di testo

The position of an object moving vertically along a line is given by the function s(t)=−4.9t2+30t+20s\(\left\)(t\(\right\))=-4.9t^2+30t+20. Find the average velocity of the object over the following intervals.

[0,h]\(\left\[\lbrack\)0,h\(\right\]\rbrack\), where h>0h\(\gt{0}\) is a real number

415
views
Domanda del libro di testo

The position of an object moving vertically along a line is given by the function s(t)=−16t2+128ts\(\left\)(t\(\right\))=-16t^2+128t. Find the average velocity of the object over the following intervals.

[1,2]\(\left\[\lbrack\)1,2\(\right\]\rbrack\)

397
views