1. The problem asks for the instantaneous rate of change of the function \( g(x) = \lim_{h \to 0} \frac{\sin(x+h) - \sin x}{h} \) at \( x = \frac{\pi}{3} \).\n\n2. Recognize that \( g(x) \) is the definition of the derivative of \( \sin x \), so \( g(x) = \frac{d}{dx} \sin x = \cos x \).\n\n3. To find the instantaneous rate of change at \( x = \frac{\pi}{3} \), evaluate \( g\left(\frac{\pi}{3}\right) = \cos\left(\frac{\pi}{3}\right) \).\n\n4. Recall that \( \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \).\n\n5. Therefore, the instantaneous rate of change of \( g \) with respect to \( x \) at \( x = \frac{\pi}{3} \) is \( \frac{1}{2} \).\n\nFinal answer: C \( \frac{1}{2} \)