1. **State the problem:** Evaluate the limit as $r$ approaches 0 of the expression $$\frac{5r^{-0.4} - 7r^{-\frac{3}{5}} + 3r^{-0.1} - 4r^{-\frac{3}{7}} - 8r^{-0.2}}{}$$ 2. **Rewrite the expression clearly:** The expression is $$5r^{-0.4} - 7r^{-\frac{3}{5}} - 3 + 3r^{-0.1} - 4r^{-\frac{3}{7}} - 8r^{-0.2}$$ (assuming the minus signs are separating terms; the problem is ambiguous but we interpret as a sum of terms with powers of $r$) 3. **Analyze each term as $r \to 0$:** - For $r^a$ with negative exponent $a$, $r^a = \frac{1}{r^{-a}}$ which tends to $\infty$ as $r \to 0^+$. - Specifically, terms like $r^{-0.4}$, $r^{-\frac{3}{5}}$, $r^{-0.1}$, $r^{-\frac{3}{7}}$, and $r^{-0.2}$ all tend to $+\infty$ as $r \to 0^+$. 4. **Dominant term:** The term with the largest negative exponent dominates. Compare exponents: - $-0.4 = -\frac{2}{5} = -0.4$ - $-\frac{3}{5} = -0.6$ - $-0.1$ - $-\frac{3}{7} \approx -0.4286$ - $-0.2$ The most negative exponent is $-0.6$ from $7r^{-\frac{3}{5}}$. 5. **Limit behavior:** Since $7r^{-\frac{3}{5}}$ dominates and is multiplied by $-7$, the term tends to $-\infty$ as $r \to 0^+$. 6. **Conclusion:** The limit diverges to $-\infty$. **Final answer:** $$\lim_{r \to 0} \left(5r^{-0.4} - 7r^{-\frac{3}{5}} - 3 + 3r^{-0.1} - 4r^{-\frac{3}{7}} - 8r^{-0.2}\right) = -\infty$$