1. **Problem:** Find the limit $$\lim_{x \to 0} \frac{\sin(2x)}{2x}$$. **Step 1:** Recall the standard limit formula $$\lim_{x \to 0} \frac{\sin x}{x} = 1$$. **Step 2:** Substitute $u = 2x$, so as $x \to 0$, $u \to 0$. **Step 3:** Rewrite the limit as $$\lim_{u \to 0} \frac{\sin u}{u}$$. **Step 4:** Using the standard limit, this equals 1. **Answer:** $$\lim_{x \to 0} \frac{\sin(2x)}{2x} = 1$$.