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: \( \lim_{x \to \infty} (\sqrt{x^2 + x} - \sqrt{x^2 - x}) \).
2. To simplify the expression, multiply and divide by the conjugate: \( \frac{(\sqrt{x^2 + x} - \sqrt{x^2 - x})(\sqrt{x^2 + x} + \sqrt{x^2 - x})}{\sqrt{x^2 + x} + \sqrt{x^2 - x}} \).
3. The numerator becomes a difference of squares: \((x^2 + x) - (x^2 - x) = 2x\).
4. The expression simplifies to \( \frac{2x}{\sqrt{x^2 + x} + \sqrt{x^2 - x}} \).
5. 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 \(x \to \infty\).

Key Concepts

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.

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.

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.