Determine the following limits.lim⁡θ→π2+13tan⁡θ{\displaystyle\lim_{\theta\to\frac{\pi}{2}^{+}}\frac13\tan\theta}

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 testo

Briggs 3rd Edition

Ch. 2 - Limits

Problema 2.4.42

Capitolo 2, Problema 2.4.42

Determine the following limits.

$\lim_{θ \to {\frac{π}{2}}^{+}} \frac{1}{3} \tan θ$

Guida verificata passo dopo passo

Step 1: Understand the problem. We need to find the limit of the function $ \frac{1}{3} \tan(\theta) $ as $ \theta $ approaches $ \frac{\pi}{2}^+ $. This means we are considering values of $ \theta $ that are slightly greater than $ \frac{\pi}{2} $.

Step 2: Analyze the behavior of $ \tan(\theta) $ near $ \frac{\pi}{2} $. The tangent function, $ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} $, becomes undefined at $ \theta = \frac{\pi}{2} $ because $ \cos(\theta) = 0 $ at this point.

Step 3: Consider the right-hand limit. As $ \theta $ approaches $ \frac{\pi}{2}^+ $, $ \cos(\theta) $ approaches 0 from the negative side, making $ \tan(\theta) $ approach $-\infty $.

Step 4: Apply the limit to the function. Since $ \tan(\theta) \to -\infty $ as $ \theta \to \frac{\pi}{2}^+ $, the expression $ \frac{1}{3} \tan(\theta) $ will also approach $-\infty $.

Step 5: Conclude the limit. Therefore, the limit $ \lim_{\theta \to \frac{\pi}{2}^+} \frac{1}{3} \tan(\theta) = -\infty $.

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:

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 how functions behave near points of interest, including points where they may not be defined. In this case, we are examining the limit of the tangent function as the angle approaches π/2 from the right, which is crucial for determining the function's behavior near that point.

Video consigliato:

05:50

One-Sided Limits

Tangent Function

The tangent function, denoted as tan(θ), is a trigonometric function defined as the ratio of the opposite side to the adjacent side in a right triangle. It can also be expressed as sin(θ)/cos(θ). As θ approaches π/2, the cosine of θ approaches zero, causing the tangent function to approach infinity, which is essential for evaluating the limit in the given question.

Video consigliato:

Percorso guidato

05:13

Slopes of Tangent Lines

One-Sided Limits

One-sided limits refer to the value that a function approaches as the input approaches a specific point from one side only, either from the left or the right. In this problem, we are interested in the limit as θ approaches π/2 from the right (denoted as θ → π/2⁺), which is important because the behavior of the tangent function differs when approaching from different directions, particularly near vertical asymptotes.

Video consigliato:

05:50

One-Sided Limits

Pratica correlata

Domanda del libro di testo

Use the precise definition of infinite limits to prove the following limits.

$\lim_{x \to - 1} \frac{1}{{(x + 1)}^{4}} = \infty$