1. **State the problem:** Evaluate the limit $$\lim_{h \to 0} \frac{\sin\left(\frac{\pi}{3} + h\right) - \sin\left(\frac{\pi}{3}\right)}{h}$$. 2. **Recall the formula:** This limit is the definition of the derivative of the function $f(x) = \sin x$ at $x = \frac{\pi}{3}$. 3. **Derivative rule:** The derivative of $\sin x$ is $\cos x$. 4. **Apply the derivative:** So, $$\lim_{h \to 0} \frac{\sin\left(\frac{\pi}{3} + h\right) - \sin\left(\frac{\pi}{3}\right)}{h} = f'\left(\frac{\pi}{3}\right) = \cos\left(\frac{\pi}{3}\right).$$ 5. **Evaluate cosine:** We know that $$\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}.$$ 6. **Final answer:** $$\boxed{\frac{1}{2}}.$$