84. Find lim(x→∞) (√(x² + 1) - √x).

검증된 단계별 안내

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

Recognize that as \(x\) approaches infinity, both \(\sqrt{x^{2} + 1}\) and \(\sqrt{x}\) grow large, so direct substitution leads to an indeterminate form of type \(\infty - \infty\).

To resolve this, multiply and divide the expression by the conjugate of the difference to rationalize it: multiply by \(\frac{\sqrt{x^{2} + 1} + \sqrt{x}}{\sqrt{x^{2} + 1} + \sqrt{x}}\).

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

Rewrite the limit as \(\lim_{x \to \infty} \frac{(x^{2} + 1) - x}{\sqrt{x^{2} + 1} + \sqrt{x}}\) and then simplify the numerator and denominator separately to analyze the behavior as \(x\) approaches infinity.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Limits at Infinity

Limits at infinity describe the behavior of a function as the input grows without bound. Understanding how expressions behave as x approaches infinity helps determine if the function approaches a finite value, infinity, or does not exist.

Rationalizing Expressions

Rationalizing involves multiplying by a conjugate to simplify expressions with square roots. This technique helps eliminate radicals in the numerator or denominator, making it easier to evaluate limits or simplify complex expressions.

Asymptotic Behavior of Square Root Functions

For large x, √(x² + 1) behaves like |x| because the x² term dominates. Recognizing this helps approximate and simplify expressions involving square roots when x approaches infinity, aiding in limit evaluation.