1. **State the problem:** Find the limit $$\lim_{x \to 0} \frac{\sin x}{5x}$$. 2. **Recall the standard limit:** We know that $$\lim_{x \to 0} \frac{\sin x}{x} = 1$$. 3. **Rewrite the expression:** $$\lim_{x \to 0} \frac{\sin x}{5x} = \lim_{x \to 0} \frac{1}{5} \cdot \frac{\sin x}{x}$$ 4. **Use limit properties:** Since $$\frac{1}{5}$$ is constant, it can be factored out: $$= \frac{1}{5} \cdot \lim_{x \to 0} \frac{\sin x}{x}$$ 5. **Evaluate the limit:** Using the standard limit, $$= \frac{1}{5} \cdot 1 = \frac{1}{5}$$ 6. **Final answer:** $$\boxed{\frac{1}{5}}$$