Let $$f(x)$$ be a differentiable function at $$x = a$$ with $$f'(a) = 2$$ and $$f(a) = 4$$. Then $$\lim_{x \to a} \frac{xf(a) - af(x)}{x - a}$$ equals:

We need to evaluate $$\displaystyle\lim_{x \to a} \dfrac{x f(a) - a f(x)}{x - a}$$, given that $$f'(a) = 2$$ and $$f(a) = 4$$.

We rewrite the numerator by adding and subtracting $$af(a)$$: $$xf(a) - af(x) = f(a)(x - a) - a\big(f(x) - f(a)\big)$$.

Therefore $$\dfrac{xf(a) - af(x)}{x - a} = f(a) - a \cdot \dfrac{f(x) - f(a)}{x - a}$$.

Taking the limit as $$x \to a$$: $$\displaystyle\lim_{x \to a}\left[f(a) - a \cdot \dfrac{f(x) - f(a)}{x - a}\right] = f(a) - a \cdot f'(a) = 4 - 2a$$.

The answer is $$4 - 2a$$.