Limits, Continuity & Differentiability
Differentiability
Grade 12
Question:
<p>Suppose \( f(x) \) is differentiable at \( x = 1 \) and \( \lim_{h \to 0} \frac{1}{h} f(1+h) = 5 \), then \( f'(1) \) equals</p>
<p>3</p>
<p>4</p>
<p>5</p>
<p>6</p>
Step-by-Step Solution
Key Concept: If f is differentiable at x=1, then f(1)=0 must hold (otherwise the limit would diverge). The given limit then directly yields f'(1) by recognizing the definition of derivative.
<p><strong>Step 1:</strong> Analyze the given limit: lim(h→0) [f(1+h)/h] = 5</p><p>For this limit to exist and be finite, the numerator must approach 0 as h→0. Therefore: lim(h→0) f(1+h) = 0, which means f(1) = 0.</p><p><strong>Step 2:</strong> Rewrite the limit using the definition of derivative:</p><p>lim(h→0) [f(1+h)/h] = lim(h→0) [f(1+h) - f(1)]/h = f'(1)</p><p>This is the standard derivative definition with f(1)=0 confirmed.</p><p><strong>Step 3:</strong> From the given condition: f'(1) = 5</p><p>∴ Answer: <strong>5</strong></p>
Correct Answer: C