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 2.6.86

Capitolo 2, Problema 2.6.86

Finding Limits of Differences When x → ±∞

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

lim x → −∞ (√(x² + 3) + x)

Guida verificata passo dopo passo

Identify the expression for which you need to find the limit: \( \lim_{x \to -\infty} (\sqrt{x^2 + 3} + x) \).

To simplify the expression, multiply and divide by the conjugate: \( \frac{(\sqrt{x^2 + 3} + x)(\sqrt{x^2 + 3} - x)}{\sqrt{x^2 + 3} - x} \).

The numerator becomes a difference of squares: \((\sqrt{x^2 + 3})^2 - x^2 = x^2 + 3 - x^2 = 3\).

Now, the expression simplifies to: \( \frac{3}{\sqrt{x^2 + 3} - x} \).

Analyze the behavior of the denominator as \( x \to -\infty \). Simplify \( \sqrt{x^2 + 3} \approx |x| \) for large \( |x| \), and consider the limit of the simplified expression.

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 at Infinity

Limits at infinity involve evaluating the behavior of a function as the variable approaches positive or negative infinity. This concept is crucial for understanding how functions behave in extreme cases, often revealing horizontal asymptotes or end behavior. In the context of the given limit, we analyze how the expression behaves as x becomes very large or very small.

Video consigliato:

03:07

Cases Where Limits Do Not Exist

Conjugates in Limits

Using conjugates is a technique in calculus that simplifies expressions involving square roots. By multiplying and dividing by the conjugate, we can eliminate square roots in the numerator or denominator, making it easier to evaluate limits. This method is particularly useful when dealing with indeterminate forms that arise in limit calculations.

Video consigliato:

06:13

Limits of Rational Functions with Radicals

Square Root Functions

Square root functions, such as √(x² + 3), are essential in calculus for understanding how they behave as x approaches infinity or negative infinity. The square root of a squared term dominates the behavior of the function, allowing us to simplify the limit. Recognizing how these functions interact with linear terms is key to solving the limit problem presented.

Video consigliato:

Percorso guidato

7:24

Multiplying & Dividing Functions

Pratica correlata

Domanda del libro di testo

Using the Formal Definition

Prove the limit statements in Exercises 37–50.

limx →4 (9 − x) = 5

278

views

Domanda del libro di testo

At what points are the functions in Exercises 13–30 continuous?

y = 1/(x – 2) – 3x

235

views

Domanda del libro di testo

Limits and Infinity

Find the limits in Exercises 37–46.

x²/³ + x⁻¹

lim --------------------

x→∞ x²/³ + cos²x

236

views

Domanda del libro di testo

Finding Limits

In Exercises 3–8, find the limit of each function (a) as x → ∞ and (b) as x → −∞. (You may wish to visualize your answer with a graphing calculator or computer.)

f(x) = 2/x − 3

314

views

Domanda del libro di testo

Using the Formal Definition

Prove the limit statements in Exercises 37–50.

limx→3 (3x − 7) = 2

259

views

Domanda del libro di testo

Infinite Limits

Find the limits in Exercises 37–48. Write ∞ or −∞ where appropriate.