<p>If \(G(x) = -\sqrt{25 - x^2}\) then \(\lim_{x \to 1} \frac{G(x) - G(1)}{x - 1} =\) ______</p>
Step-by-Step Solution
Key Concept: This limit is the definition of the derivative of G(x) at x=1. Use the derivative formula for √u where u = 25-x², then evaluate at x=1.
<p><strong>Step 1:</strong> Recognize that <span style='color:blue'>lim(x→1) [G(x)-G(1)]/(x-1) = G'(1)</span> (definition of derivative)</p><p><strong>Step 2:</strong> Find G'(x) using the chain rule:<br/>G(x) = -√(25-x²)<br/>G'(x) = -1/(2√(25-x²)) · (-2x) = x/√(25-x²)</p><p><strong>Step 3:</strong> Evaluate G'(1):<br/>G'(1) = 1/√(25-1²) = 1/√(25-1) = 1/√24</p><p><strong>Simplification:</strong> 1/√24 = 1/(2√6) (rationalize if needed)</p><p>∴ Answer: <strong>1/√24</strong> or equivalently <strong>√24/24 = √6/12</strong></p>
Correct Answer: 1/√24