1. The problem is to evaluate the limit $$\lim_{h \to 0} \frac{\sin\left(\frac{\pi}{3} + h\right) - \sin\left(\frac{\pi}{3}\right)}{h}$$ which represents the derivative of $\sin x$ at $x = \frac{\pi}{3}$.\n\n2. Recall the definition of the derivative: $$f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$$ and the derivative of $\sin x$ is $\cos x$.\n\n3. Using this, the limit equals $\cos\left(\frac{\pi}{3}\right)$.\n\n4. Evaluate $\cos\left(\frac{\pi}{3}\right)$: $$\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}$$\n\n5. Therefore, the value of the limit is $$\boxed{\frac{1}{2}}$$.