Ch. 2 - Limits and Continuity

Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo

Hass 15th Edition

Ch. 2 - Limits and Continuity

Problema 59a

Capitolo 2, Problema 59a

Find the limits in Exercises 59–62. Write ∞ or −∞ where appropriate.

lim (2 − 3 / t¹/³) as

a. t → 0⁺

Guida verificata passo dopo passo

Identify the expression for which you need to find the limit: \( 2 - \frac{3}{t^{1/3}} \).

Recognize that as \( t \to 0^+ \), \( t^{1/3} \to 0^+ \) as well, since the cube root of a positive number approaching zero also approaches zero.

Consider the behavior of the term \( \frac{3}{t^{1/3}} \). As \( t^{1/3} \to 0^+ \), \( \frac{3}{t^{1/3}} \to +\infty \) because dividing by a very small positive number results in a very large positive number.

Analyze the entire expression \( 2 - \frac{3}{t^{1/3}} \). Since \( \frac{3}{t^{1/3}} \to +\infty \), the expression \( 2 - \frac{3}{t^{1/3}} \) will tend towards \(-\infty\).

Conclude that the limit of the expression as \( t \to 0^+ \) is \(-\infty\).

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 specific points, including points of discontinuity or infinity. In this case, we are interested in the limit of the function as t approaches 0 from the positive side (0⁺).

Video consigliato:

05:50

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 the left (−) or the right (+). In this question, we are evaluating the right-hand limit as t approaches 0, which is crucial for determining the function's behavior in that region without considering values from the left.

Video consigliato:

05:50

One-Sided Limits

Cube Root Function

The cube root function, denoted as t¹/³, is a continuous function that returns the number which, when cubed, gives the input value. As t approaches 0, the cube root of t also approaches 0. Understanding how this function behaves near 0 is essential for evaluating the limit in the given problem.

Video consigliato:

Percorso guidato

5:57

Graphs of Common Functions

Pratica correlata

Domanda del libro di testo

Find the limits in Exercises 53–58. Write ∞ or −∞ where appropriate.

lim (x² − 3x + 2) / (x³ − 4x) as

b. x→−2⁺

190

views

Domanda del libro di testo

Find the limits in Exercises 53–58. Write ∞ or −∞ where appropriate.

lim (x² − 3x + 2) / (x³ − 2x²) as

a. x→0⁺

184

views

Domanda del libro di testo

Theory and Examples

Suppose that g(x) ≤ f(x) ≤ h(x) for all x≠2 and suppose that lim x→2 g(x) = lim x→2 h(x) = −5. Can we conclude anything about the values of f, g, and h at x = 2? Could f(2) = 0? Could limx→2 f(x)=0? Give reasons for your answers.

310

views

Domanda del libro di testo

Theory and Examples

If x⁴ ≤ f(x) ≤ x² for x in [−1,1] and x² ≤ f(x) ≤ x⁴ for x < - 1 and x > 1, at what points c do you automatically know limx→c f(x)? What can you say about the value of the limit at these points?

320

views

Domanda del libro di testo

Use the formal definitions from Exercise 97 to prove the limit statements in Exercises 98–102.

lim x→2⁻ (1 / (x − 2)) = −∞

285

views

Domanda del libro di testo

Find the limits in Exercises 53–58. Write ∞ or −∞ where appropriate.

lim (x² − 3x + 2) / (x³ − 2x²) as

d. x→2

207