Continuity and Differentiability
Differentiability definition
Premium Question
Grade 12
Question:
If $f$ is differentiable at $x = a$, then $\lim_{x \to a} \frac{xf(a) - af(x)}{x - a}$ equals:
(1) $f(a) - af'(a)$
(2) $af'(a) - f(a)$
(3) $f(a) + af'(a)$
(4) $f'(a)$
Step-by-Step Solution
Key Concept: Write $xf(a)-af(x) = xf(a)-af(a)+af(a)-af(x) = (x-a)f(a)-a(f(x)-f(a))$. Divide by $(x - a)$: $f(a) - a \cdot \frac{f(x)-f(a)}{x-a} \to f(a) - af'(a)$.
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Correct Answer: (1)