Skip to main content
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 testoBriggs 3rd EditionCh. 2 - LimitsProblema 2.R.41
Capitolo 2, Problema 2.R.41

Determine the following limits.
lim x→∞ (3 tan-1 x + 2)

Guida verificata passo dopo passo
1
Recognize that the problem involves finding the limit of a function as \( x \) approaches infinity: \( \lim_{x \to \infty} (3 \tan^{-1} x + 2) \).
Recall the behavior of the inverse tangent function, \( \tan^{-1} x \), as \( x \to \infty \). The function approaches \( \frac{\pi}{2} \).
Substitute the asymptotic value of \( \tan^{-1} x \) into the expression: \( 3 \tan^{-1} x + 2 \approx 3 \left( \frac{\pi}{2} \right) + 2 \).
Simplify the expression by multiplying and adding the constants: \( 3 \times \frac{\pi}{2} + 2 \).
Conclude that the limit is the simplified expression from the previous step.

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 at Infinity

Limits at infinity involve evaluating the behavior of a function as the input approaches infinity. This concept is crucial for understanding how functions behave in extreme cases, allowing us to determine their end behavior. In this context, we analyze how the function approaches a specific value as x becomes very large.
Video consigliato:
05:50
One-Sided Limits

Inverse Tangent Function

The inverse tangent function, denoted as tan<sup>-1</sup>(x) or arctan(x), is a function that returns the angle whose tangent is x. As x approaches infinity, the value of arctan(x) approaches π/2. This property is essential for solving the limit in the question, as it helps us understand the limiting behavior of the function involved.
Video consigliato:
3:17
Inverse Tangent

Constant Addition in Limits

When evaluating limits, adding a constant to a function does not affect the limit itself. This principle allows us to simplify the limit calculation by focusing on the behavior of the variable part of the function. In this case, after determining the limit of the arctan function, we can simply add 2 to find the final limit value.
Video consigliato:
05:50
One-Sided Limits
Pratica correlata
Domanda del libro di testo

The height above the ground of a stone thrown upwards is given by s(t), where t is measured in seconds. After 1 second, the height of the stone is 48 feet above the ground, and after 1.5 seconds, the height of the stone is 60 feet above the ground. Evaluate s(1) and s(1.5), and then find the average velocity of the stone over the time interval [1, 1.5].

252
views
Domanda del libro di testo

b. Estimate a solution to the equation in the given interval using a root finder.


x=cos x; (0,π/2)

215
views
Domanda del libro di testo

Find the intervals on which the following functions are continuous. Specify right- or left-continuity at the finite endpoints.

g(x)=cos(ex)g\(\left\)(x\(\right\))=\(\cos\)\(\left\)(e^{x}\(\right\))

252
views
Domanda del libro di testo

Determine the following limits.

lim x→∞ (2x − 3) / (4x + 10)

395
views
Domanda del libro di testo

Use the graph of ff in the figure to determine the values of xx in the interval (−3,5)\(\left\)(-3,5\(\right\)) at which f fails to be continuous. Justify your answers using the continuity checklist.

<IMAGE>

314
views
Domanda del libro di testo

Let g(x)={5x−2if x<1aif x=1ax2+bxif x>1g\(\left\)(x\(\right\))=\(\begin{cases}\)5x-2 & \(\text{if }\)x<1\\ a & \(\text{if }\)x=1\\ ax^2+bx & \(\text{if }\)x>1\(\end{cases}\).


Determine values of the constants aa and bb , if possible, for which gg is continuous at x=1x=1.

361
views