1. The problem is to evaluate the limit: $$\lim_{x \to 0} \frac{du(x+h) - du(h)}{x}$$ 2. This expression resembles the definition of the derivative of the function $u$ evaluated at $h$. 3. Recall that the derivative $u'(h)$ is defined as: $$u'(h) = \lim_{x \to 0} \frac{u(h+x) - u(h)}{x}$$ 4. If $du$ represents $u$, then: $$\lim_{x \to 0} \frac{du(x+h) - du(h)}{x} = u'(h)$$ 5. Therefore, the limit is the derivative of $u$ at $h$. Final answer: $$\lim_{x \to 0} \frac{du(x+h) - du(h)}{x} = u'(h)$$