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Ch. 2 - Limits and Continuity
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 2 - Limits and ContinuityProblem 2.6.90
Chapter 2, Problem 2.6.90

Finding Limits of Differences When x → ±∞


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


lim x → ∞ (√(x² + x) − √(x² − x))

Verified step by step guidance
1
Identify the expression whose limit you need to find: \( \lim_{x \to \infty} (\sqrt{x^2 + x} - \sqrt{x^2 - x}) \).
To simplify the expression, multiply and divide by the conjugate: \( \frac{(\sqrt{x^2 + x} - \sqrt{x^2 - x})(\sqrt{x^2 + x} + \sqrt{x^2 - x})}{\sqrt{x^2 + x} + \sqrt{x^2 - x}} \).
The numerator becomes a difference of squares: \((x^2 + x) - (x^2 - x) = 2x\).
The expression simplifies to \( \frac{2x}{\sqrt{x^2 + x} + \sqrt{x^2 - x}} \).
Divide the numerator and the denominator by \(x\) to simplify further: \( \frac{2}{\sqrt{1 + \frac{1}{x}} + \sqrt{1 - \frac{1}{x}}} \), and evaluate the limit as \(x \to \infty\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Limits at Infinity

Limits at infinity involve finding the value that a function approaches as the input grows indefinitely large or small. This concept is crucial for understanding the behavior of functions as x approaches positive or negative infinity, often simplifying complex expressions to determine their end behavior.
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Cases Where Limits Do Not Exist

Conjugate Multiplication

Multiplying by the conjugate is a technique used to simplify expressions, especially those involving square roots. By multiplying the numerator and denominator by the conjugate, we can eliminate radicals, making it easier to evaluate limits and simplify expressions.
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Limits of Rational Functions with Radicals

Simplifying Radical Expressions

Simplifying radical expressions involves manipulating terms to reduce complexity, often by rationalizing denominators or combining like terms. This process is essential for evaluating limits, as it allows for clearer insight into the behavior of functions involving square roots or other radicals.
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Simplifying Trig Expressions
Related Practice
Textbook Question

Graphing Simple Rational Functions


Graph the rational functions in Exercises 63–68. Include the graphs and equations of the asymptotes and dominant terms.


y = −3/(x − 3)

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Textbook Question

Limits with trigonometric functions


Find the limits in Exercises 43–50.


limx→0 (1 + x + sin x) / (3 cosx)

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Textbook Question

Exercises 5–10 refer to the function

f(x) = { x² − 1, −1 ≤ x < 0

2x, 0 < x < 1

1, x = 1

−2x + 4, 1 < x < 2

0, 2 < x < 3

graphed in the accompanying figure.

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At what values of x is f continuous?

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Textbook Question

Use formal definitions to prove the limit statements in Exercises 93–96.


lim x → 0 (1 / |x|) = ∞

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Textbook Question

Graphing Simple Rational Functions


Graph the rational functions in Exercises 63–68. Include the graphs and equations of the asymptotes and dominant terms.


y = 1/(x − 1)

348
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Textbook Question

Using the Formal Definitions


Use the formal definitions of limits as x → ±∞ to establish the limits in Exercises 91 and 92.


If f has the constant value f(x) = k, then lim x → ∞ f(x) = k.

315
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