Limits, Continuity & Differentiability
Differentiability and derivatives
Grade 12
<p>If \(\displaystyle\lim_{h \to 0} \frac{1}{h} f(1+h) = 5\), find \(f'(1)\).</p>
Step-by-Step Solution
Key Concept: Recognize that the given limit is NOT the standard derivative definition. Rewrite it as lim(h→0) [f(1+h) - f(1)]/h by recognizing that if this equals 5, then f(1) must equal 0 (from the necessary condition that f must be continuous at the derivative point).
<p><strong>Step 1:</strong> Analyze the given limit form: lim(h→0) [f(1+h)]/h = 5</p><p><strong>Step 2:</strong> For this limit to exist and be finite, f(1+h) must approach 0 as h→0, which means f(1) = 0 (by continuity at x=1).</p><p><strong>Step 3:</strong> Rewrite the limit: lim(h→0) [f(1+h)]/h = lim(h→0) [f(1+h) - f(1)]/h = lim(h→0) [f(1+h) - 0]/h</p><p><strong>Step 4:</strong> This is precisely the definition of f'(1) = lim(h→0) [f(1+h) - f(1)]/h</p><p><strong>Step 5:</strong> Therefore, f'(1) = 5</p><p>∴ Answer: C</p>
Correct Answer: C