1. **Stating the problem:** We are given several mathematical expressions and asked to list them from lowest value to greatest value. 2. **Evaluate each expression:** - $\sum\limits_{i=2}^3 i = 2 + 3 = 5$ - $e^3 \approx 20.0855$ - $6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720$ - $\int_3^6 x \, dx = \left[\frac{x^2}{2}\right]_3^6 = \frac{6^2}{2} - \frac{3^2}{2} = \frac{36}{2} - \frac{9}{2} = \frac{27}{2} = 13.5$ - $\sqrt{9} = 3$ - $\frac{6\pi}{2} = 3\pi \approx 3 \times 3.1416 = 9.4248$ - $\log_4(25) = \frac{\log_{10}(25)}{\log_{10}(4)} \approx \frac{1.39794}{0.60206} \approx 2.322$ - $\frac{3}{9} = \frac{1}{3} \approx 0.3333$ - $\infty$ (infinity) is the greatest value. 3. **Sorting from lowest to greatest:** $$0.3333 < 3 < 5 < 9.4248 < 13.5 < 20.0855 < 720 < \infty$$ Including all values with their expressions: $$\frac{3}{9} < \sqrt{9} < \sum_{i=2}^3 i < \frac{6\pi}{2} < \int_3^6 x \, dx < e^3 < 6! < \infty$$