<p>$$\lim_{x \to a^+} \frac{x - b - \sqrt{a - b}}{\sqrt{x^2 - a^2}}$$, $$(a > b)$$ is</p>
Step-by-Step Solution
Key Concept: As x → a⁺, the denominator √(x² - a²) → 0⁺, so we must analyze the numerator's behavior to determine if the limit exists and is finite. The numerator approaches (a - b - √(a-b)), which is generally non-zero, creating an indeterminate form that requires careful evaluation.
<p><strong>Step 1:</strong> Analyze the denominator behavior. As x → a⁺, we have x² → a², so √(x² - a²) → 0⁺.</p><p><strong>Step 2:</strong> Analyze the numerator behavior. As x → a⁺, the numerator approaches: a - b - √(a - b).</p><p><strong>Step 3:</strong> For a > b, we have a - b > 0, so √(a - b) exists and is positive. The sign of (a - b - √(a - b)) depends on whether a - b is greater than, equal to, or less than √(a - b).</p><p><strong>Step 4:</strong> Let u = a - b > 0. We need to examine the sign of u - √u. For u > 1: u - √u > 0. For 0 < u < 1: u - √u < 0. For u = 1: u - √u = 0.</p><p><strong>Step 5:</strong> Since the numerator has an indeterminate sign depending on the relationship between a and b, and the denominator → 0⁺, the limit either diverges to +∞, -∞, or is undefined without specifying the exact values of a and b.</p><p><strong>Step 6:</strong> The problem statement does not provide specific values for a and b beyond a > b. Therefore, the limit cannot be uniquely specified without additional constraints on the relationship between a and b.</p><p><strong>∴ Answer:</strong> Not specified</p>
Correct Answer: Not specified