Skip to main content
Ch. 2 - Limits and Continuity
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. 2 - Limits and ContinuityProblema 2.2.47
Capitolo 2, Problema 2.2.47

Limits with trigonometric functions


Find the limits in Exercises 43–50.


lim x→0 tan x

Guida verificata passo dopo passo
1
Recognize that the limit involves the trigonometric function tan(x) as x approaches 0.
Recall the identity for tangent: tan(x) = sin(x) / cos(x). This will help in simplifying the expression.
Consider the limit: lim x→0 tan(x) = lim x→0 (sin(x) / cos(x)).
Use the fact that as x approaches 0, sin(x) approaches 0 and cos(x) approaches 1. This simplifies the expression.
Apply the limit: lim x→0 (sin(x) / cos(x)) = (lim x→0 sin(x)) / (lim x→0 cos(x)) = 0 / 1.

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:
2m

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 where the function is not explicitly defined. Evaluating limits is essential for defining derivatives and integrals.
Video consigliato:
05:50
One-Sided Limits

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, relate angles to ratios of sides in right triangles. They are periodic functions that play a crucial role in calculus, especially when dealing with limits, derivatives, and integrals involving angles. Understanding their properties and behaviors is key to solving limit problems involving these functions.
Video consigliato:
Percorso guidato
6:04
Introduction to Trigonometric Functions

L'Hôpital's Rule

L'Hôpital's Rule is a method used to evaluate 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 when dealing with limits involving trigonometric functions.
Video consigliato:
5:50
Power Rules
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
1
rank
Domanda del libro di testo

Finding Limits of Differences When x → ±∞


Find the limits in Exercises 84–90. (Hint: Try multiplying and dividing by the conjugate.)


lim x → −∞ (2x + √(4x² + 3x − 2))

289
views
Domanda del libro di testo

Using the Sandwich Theorem


If 2−x² ≤ g(x) ≤ 2cosx for all x, find limx→0 g(x).

364
views
Domanda del libro di testo

Slope of a Curve at a Point


In Exercises 7–18, use the method in Example 3 to find (a) the slope of the curve at the given point P, and (b) an equation of the tangent line at P.


y=√7−x, P(−2,3)

325
views
Domanda del libro di testo

Using limθ→0 sin θ / θ = 1


Find the limits in Exercises 23–46.


limθ→0 sin θ / sin 2θ

308
views
Domanda del libro di testo

Finding Deltas Algebraically


Each of Exercises 15–30 gives a function f(x) and numbers L, c, and ε>0. In each case, find the largest open interval about c on which the inequality |f(x)−L| <ε holds. Then give a value for δ>0 such that for all x satisfying 0 < |x−c| < δ, the inequality |f(x)−L| < ε holds.


f(x) = 1/x, L = 1/4, c = 4, ε = 0.05

313
views