Limits, Continuity & Differentiability
Limits and Derivatives
Grade 12
Question:
<p>Let \(f(x)\) be a continuous and differentiable function such that \(\displaystyle\lim_{h \to 0} \frac{f(3+7h) - f(3+4h)}{h} = 4\). Then the value of \(f'(3)\) equals:</p>
<p>(a) \(\dfrac{1}{3}\)</p>
<p>(b) \(\dfrac{2}{3}\)</p>
<p>(c) \(\dfrac{4}{3}\)</p>
<p>(d) \(1\)</p>
Step-by-Step Solution
Key Concept: Use the definition of derivative by rewriting the given limit expression in terms of f'(3). Factor out coefficients strategically: lim[h→0] [f(3+7h)-f(3+4h)]/h = lim[h→0] [7·(f(3+7h)-f(3))/7h - 4·(f(3+4h)-f(3))/4h], which becomes 7f'(3) - 4f'(3) = 3f'(3).
<p><strong>Step 1:</strong> Rewrite the limit by splitting the numerator into two parts:</p><p>$$\lim_{h \to 0} \frac{f(3+7h) - f(3+4h)}{h} = \lim_{h \to 0} \frac{[f(3+7h)-f(3)] - [f(3+4h)-f(3)]}{h}$$</p><p><strong>Step 2:</strong> Separate into two limits with appropriate scaling:</p><p>$$= \lim_{h \to 0} \frac{f(3+7h)-f(3)}{h} - \lim_{h \to 0} \frac{f(3+4h)-f(3)}{h}$$</p><p><strong>Step 3:</strong> Rewrite using standard derivative form by multiplying and dividing:</p><p>$$= \lim_{h \to 0} 7 \cdot \frac{f(3+7h)-f(3)}{7h} - \lim_{h \to 0} 4 \cdot \frac{f(3+4h)-f(3)}{4h}$$</p><p><strong>Step 4:</strong> Apply the definition of derivative. As h→0, we have 7h→0 and 4h→0:</p><p>$$= 7 \cdot f'(3) - 4 \cdot f'(3) = 3f'(3)$$</p><p><strong>Step 5:</strong> Set equal to the given limit value:</p><p>$$3f'(3) = 4$$</p><p>$$f'(3) = \frac{4}{3}$$</p><p>∴ Answer: C</p>
Correct Answer: C