Use l’Hôpital’s Rule to find the limits in Exercises 85–108.
95. lim(x→∞) (√(x² + x + 1) - √(x² - x))

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Identify the limit expression: \(\lim_{x \to \infty} \left( \sqrt{x^{2} + x + 1} - \sqrt{x^{2} - x} \right)\).

Recognize that as \(x \to \infty\), both \(\sqrt{x^{2} + x + 1}\) and \(\sqrt{x^{2} - x}\) behave like \(x\), so the expression is of the indeterminate form \(\infty - \infty\).

To apply l’Hôpital’s Rule, first rewrite the expression to a quotient form that yields an indeterminate form \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\). Multiply and divide by the conjugate to rationalize the expression: multiply numerator and denominator by \(\sqrt{x^{2} + x + 1} + \sqrt{x^{2} - x}\).

Simplify the numerator using the difference of squares formula: \((a - b)(a + b) = a^{2} - b^{2}\), which gives \(\left( x^{2} + x + 1 \right) - \left( x^{2} - x \right) = 2x + 1\).

Rewrite the limit as \(\lim_{x \to \infty} \frac{2x + 1}{\sqrt{x^{2} + x + 1} + \sqrt{x^{2} - x}}\) and then analyze this new expression to find the limit, possibly applying l’Hôpital’s Rule if the form is still indeterminate.

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Limits at Infinity

Limits at infinity describe the behavior of a function as the input grows without bound. Understanding how functions behave as x approaches infinity helps determine if the function approaches a finite value, infinity, or does not settle. This concept is essential for evaluating the given limit as x approaches infinity.

l’Hôpital’s Rule

l’Hôpital’s Rule is a method for evaluating limits that result in indeterminate forms like 0/0 or ∞/∞. It involves differentiating the numerator and denominator separately and then taking the limit of their quotient. This rule simplifies complex limits, especially those involving roots or polynomials.

Algebraic Manipulation of Radicals

Algebraic manipulation of radicals involves techniques like rationalizing expressions to simplify limits involving square roots. For example, multiplying by the conjugate can eliminate radicals in the numerator or denominator, making it easier to apply l’Hôpital’s Rule or directly evaluate the limit.

Use l’Hôpital’s Rule to find the limits in Exercises 85–108.95. lim(x→∞) (√(x² + x + 1) - √(x² - x))