Skip to main content
Ch. 4 - Applications of the Derivative
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. 4 - Applications of the DerivativeProblema 4.7.33
Capitolo 4, Problema 4.7.33

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.


lim_x→ 0 (1 - cos 3x) / 8x²

Guida verificata passo dopo passo
1
First, identify the form of the limit as x approaches 0. Substitute x = 0 into the expression (1 - cos(3x)) / 8x². This results in the indeterminate form 0/0, which suggests that l'Hôpital's Rule may be applicable.
Apply l'Hôpital's Rule, which states that if the limit of f(x)/g(x) as x approaches a value results in an indeterminate form 0/0 or ∞/∞, then the limit can be found by differentiating the numerator and the denominator separately. Differentiate the numerator: the derivative of 1 - cos(3x) is 3sin(3x).
Differentiate the denominator: the derivative of 8x² is 16x.
Rewrite the limit using the derivatives: lim_(x→0) (3sin(3x)) / (16x).
Evaluate the new limit as x approaches 0. Substitute x = 0 into the expression (3sin(3x)) / (16x). Since sin(3x) approaches 0 as x approaches 0, the limit can be further simplified using the standard limit lim_(u→0) (sin(u)/u) = 1, where u = 3x. This will help in finding the final value of the limit.

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:
9m

Concetti chiave

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

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 the function's behavior near points of interest, including points where the function may not be explicitly defined. Evaluating limits is crucial for determining continuity, derivatives, and integrals.
Video consigliato:
05:50
One-Sided Limits

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 separately. This rule simplifies the process of finding limits in complex functions.
Video consigliato:
5:50
Power Rules

Trigonometric Limits

Trigonometric limits involve evaluating limits that include trigonometric functions, such as sine and cosine. A common limit is lim_x→0 (1 - cos(x))/x², which approaches 0. Understanding the behavior of trigonometric functions near specific points is essential for solving limits involving these functions, especially when applying L'Hôpital's Rule.
Video consigliato:
Percorso guidato
6:04
Introduction to Trigonometric Functions
Pratica correlata
Domanda del libro di testo

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.


lim_x→π/2⁻ (π/2 - x) sec x

180
views
Domanda del libro di testo

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.


lim_x→1⁻ (1-x) tan πx/2

242
views
Domanda del libro di testo

Second Derivative Test Locate the critical points of the following functions. Then use the Second Derivative Test to determine (if possible) whether they correspond to local maxima or local minima.


f(x) = 2x⁻³ - x⁻²

196
views
Domanda del libro di testo

Use the following graphs to identify the points on the interval [a, b] at which local and absolute extreme values occur. <IMAGE>

250
views
Domanda del libro di testo

{Use of Tech} Write the formula for Newton’s method and use the given initial approximation to compute the approximations x₁ and x₂.


f(x) = e⁻ˣ - x; x₀ = ln 2

210
views
Domanda del libro di testo

23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.


∫ (4√x - (4 /√x)) dx

105
views
1
rank