1. **State the problem:** We need to find the limit $$\lim_{x \to 5} \left(\frac{1}{5+x}\right)(10+2x)$$ 2. **Recall the limit properties:** If the function is continuous at the point, we can directly substitute the value of $x$ into the function. 3. **Substitute $x=5$ into the expression:** $$\left(\frac{1}{5+5}\right)(10+2 \times 5) = \left(\frac{1}{10}\right)(10+10)$$ 4. **Simplify the expression:** $$\left(\frac{1}{10}\right)(20) = 2$$ 5. **Conclusion:** The limit is $2$. The function is continuous at $x=5$, so direct substitution is valid. Final answer: $$\boxed{2}$$