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 4.7.37
Capitolo 4, Problema 4.7.37

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


lim_x→ π/2⁻ (tanx ) / (3 / (2x - π))

Guida verificata passo dopo passo
1
Identify the form of the limit as x approaches π/2 from the left. Both the numerator tan(x) and the denominator 3/(2x - π) approach infinity, creating an indeterminate form of type ∞/∞.
Since the limit is in the indeterminate form ∞/∞, apply l'Hôpital's Rule, which allows us to differentiate the numerator and the denominator separately.
Differentiate the numerator: The derivative of tan(x) with respect to x is sec²(x).
Differentiate the denominator: The derivative of 3/(2x - π) with respect to x is -6/(2x - π)².
Re-evaluate the limit using the derivatives: lim_{x→π/2⁻} (sec²(x)) / (-6/(2x - π)²). Simplify the expression and evaluate the limit as x approaches π/2 from the left.

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:
6m

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 of discontinuity or infinity. Evaluating limits is crucial for defining derivatives and integrals, which are core components of calculus.
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

Tangent Function

The tangent function, denoted as tan(x), is a periodic function defined as the ratio of the sine and cosine functions: tan(x) = sin(x)/cos(x). It has vertical asymptotes where the cosine function is zero, which occurs at odd multiples of π/2. Understanding the behavior of the tangent function near these points is essential for evaluating limits involving tan(x).
Video consigliato:
Percorso guidato
05:13
Slopes of Tangent Lines
Pratica correlata
Domanda del libro di testo

Minimum sum Find positive numbers x and y satisfying the equation xy = 12 such that the sum 2x + y is as small as possible.

205
views
Domanda del libro di testo

If F(x) = x² - 3x + C and F (-1) = 4 , what is the value of C?

39
views
Domanda del libro di testo

Maximum-volume cone A cone is constructed by cutting a sector from a circular sheet of metal with radius 20. The cut sheet is then folded up and welded (see figure). Find the radius and height of the cone with maximum volume that can be formed in this way. <IMAGE>

530
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 (x³ - x² - 5x - 3)/(x⁴ + 2x³ - x² -4x -2)

233
views
Domanda del libro di testo

Differentials Consider the following functions and express the relationship between a small change in x and the corresponding change in y in the form dy = f'(x)dx.


f(x) = (x+4)/(4-x)

261
views
Domanda del libro di testo

13-26 Implicit differentiation Carry out the following steps.

a. Use implicit differentiation to find dy/dx.

x⁴+y⁴ = 2;(1,−1)

219
views