Finding Limits of Differences When x → ±∞


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


lim x → ∞ (√(x² + 25) − √(x² − 1))

Verified step by step guidance

1. Step 1: Recognize that the expression involves a difference of square roots, which can be challenging to evaluate directly as x approaches infinity. To simplify, consider multiplying and dividing by the conjugate of the expression.
2. Step 2: The conjugate of the expression √(x² + 25) − √(x² − 1) is √(x² + 25) + √(x² − 1). Multiply and divide the original expression by this conjugate to rationalize the numerator.
3. Step 3: After multiplying by the conjugate, the expression becomes [(√(x² + 25) − √(x² − 1)) * (√(x² + 25) + √(x² − 1))] / (√(x² + 25) + √(x² − 1)). This simplifies the numerator using the difference of squares formula: a² - b² = (a - b)(a + b).
4. Step 4: The numerator simplifies to (x² + 25) - (x² - 1), which further simplifies to 26. The expression now is 26 / (√(x² + 25) + √(x² − 1)).
5. Step 5: Evaluate the limit of the simplified expression as x approaches infinity. As x becomes very large, the dominant term in the denominator is x, so the expression approximates to 26 / (2x). The limit of this expression as x approaches infinity is 0.

Key Concepts

Limits at Infinity

Limits at infinity involve finding the behavior of a function as the variable approaches positive or negative infinity. This concept helps determine the end behavior of functions, which is crucial for understanding asymptotic behavior and horizontal asymptotes. In this context, it involves analyzing how the expression behaves as x becomes very large.

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 the square roots, making it easier to evaluate limits. This method is particularly useful in rationalizing differences of square roots, as seen in the given problem.

Simplification of Radical Expressions

Simplifying radical expressions involves manipulating expressions to make them easier to work with, often by removing radicals from the denominator or combining like terms. In the context of limits, simplification can reveal dominant terms that dictate the behavior of the function as x approaches infinity, allowing for easier limit evaluation.