Skip to main content
Ch. 7 - Transcendental Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 7 - Transcendental FunctionsProblema 7.5.51
Capitolo 7, Problema 7.5.51

Use l’Hôpital’s rule to find the limits in Exercises 7–52.
51. lim (θ → 0) (θ - sin θ cos θ) / (tan θ - θ)

Guida verificata passo dopo passo
1
First, identify the form of the limit as \( \theta \to 0 \). Substitute \( \theta = 0 \) into the numerator and denominator separately to check if the limit is an indeterminate form like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \).
Since direct substitution gives an indeterminate form \( \frac{0}{0} \), apply l'Hôpital's Rule, which states that \( \lim_{\theta \to 0} \frac{f(\theta)}{g(\theta)} = \lim_{\theta \to 0} \frac{f'(\theta)}{g'(\theta)} \) provided the latter limit exists.
Compute the derivative of the numerator: \( f(\theta) = \theta - \sin \theta \cos \theta \). Use the product rule for \( \sin \theta \cos \theta \) and the derivative of \( \theta \).
Compute the derivative of the denominator: \( g(\theta) = \tan \theta - \theta \). Use the derivative of \( \tan \theta \) and the derivative of \( \theta \).
After finding \( f'(\theta) \) and \( g'(\theta) \), write the new limit as \( \lim_{\theta \to 0} \frac{f'(\theta)}{g'(\theta)} \). Then, substitute \( \theta = 0 \) into this expression to evaluate 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:
8m

Concetti chiave

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

l’Hôpital’s Rule

l’Hôpital’s Rule is a method for evaluating limits that result in indeterminate forms like 0/0 or ∞/∞. It states that the limit of a ratio of functions can be found by taking the limit of the ratio of their derivatives, provided certain conditions are met. This rule simplifies complex limit problems involving trigonometric or other functions.
Video consigliato:
5:50
Power Rules

Trigonometric Limits and Identities

Understanding basic trigonometric limits, such as lim θ→0 (sin θ)/θ = 1 and lim θ→0 (tan θ)/θ = 1, is essential. Familiarity with trigonometric identities helps simplify expressions before or after applying l’Hôpital’s Rule, making it easier to evaluate the limit accurately.
Video consigliato:
7:17
Verifying Trig Equations as Identities

Derivative of Trigonometric Functions

Knowing how to differentiate trigonometric functions like sin θ, cos θ, and tan θ is crucial when applying l’Hôpital’s Rule. For example, the derivatives are cos θ, -sin θ, and sec² θ respectively. Correct differentiation allows the limit to be evaluated correctly after applying the rule.
Video consigliato:
06:35
Derivatives of Other Inverse Trigonometric Functions
Pratica correlata
Domanda del libro di testo

In Exercises 21–48, find the derivative of y with respect to the appropriate variable.

33. y=ln(arctan(x))

26
views
Domanda del libro di testo

In Exercises 57–70, use logarithmic differentiation to find the derivative of y with respect to the given independent variable.

61. y = √(θ + 3) sin θ

48
views
Domanda del libro di testo

In Exercises 57–70, use logarithmic differentiation to find the derivative of y with respect to the given independent variable.

64. y = 1/(t(t+1)(t+2))

30
views
Domanda del libro di testo

In Exercises 21–48, find the derivative of y with respect to the appropriate variable.

29. y=arcsec(1/t), 0<t<1

28
views
Domanda del libro di testo

In Exercises 21–48, find the derivative of y with respect to the appropriate variable.

27. y=arccsc(x²+1)

16
views
Domanda del libro di testo

Rewrite the expressions in Exercises 5–10 in terms of exponentials and simplify the results as much as you can.

8. cosh(3x) - sinh(3x)

32
views