Limits, Continuity & Differentiability
Limits
Grade 12
Question:
<p>If \(f(c) = 2\), \(f'(c) = 1\), \(g(c) = -1\), \(g'(c) = 2\) then \(\lim_{x \to c} \frac{g(x)f(c) - g(c)f(x)}{x - c} =\) ______</p>
Step-by-Step Solution
Key Concept: Recognize this limit as a derivative expression in disguise by rewriting it as a linear combination of derivatives using the definition f'(c) = lim[h→0] (f(c+h)-f(c))/h. Separate the numerator strategically by adding and subtracting g(c)f(c).
<p><strong>Step 1:</strong> Rewrite the numerator by adding and subtracting g(c)f(c):</p><p>Numerator = g(x)f(c) - g(c)f(c) + g(c)f(c) - g(c)f(x)</p><p>= f(c)[g(x) - g(c)] - g(c)[f(x) - f(c)]</p><p><strong>Step 2:</strong> Separate the limit:</p><p>$$\lim_{x \to c} \frac{f(c)[g(x) - g(c)] - g(c)[f(x) - f(c)]}{x - c}$$</p><p>$$= f(c)\lim_{x \to c}\frac{g(x) - g(c)}{x - c} - g(c)\lim_{x \to c}\frac{f(x) - f(c)}{x - c}$$</p><p><strong>Step 3:</strong> Apply derivative definition:</p><p>$$= f(c) \cdot g'(c) - g(c) \cdot f'(c)$$</p><p>$$= 2(2) - (-1)(1)$$</p><p>$$= 4 + 1 = 5$$</p><p>∴ Answer: <strong>5</strong></p>
Correct Answer: 5