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:** $$5r^{-0.4} - 7r^{-\frac{3}{5}} + 3r^{-0.1} - 4r^{-\frac{3}{7}} - 8r^{-0.2}$$ 3. **Recall the rule for limits involving powers:** For $r \to 0^+$, if the exponent is negative, $r^{\text{negative}}$ tends to $+\infty$. 4. **Evaluate each term as $r \to 0^+$:** - $5r^{-0.4} = 5 \times \frac{1}{r^{0.4}} \to +\infty$ - $-7r^{-\frac{3}{5}} = -7 \times \frac{1}{r^{0.6}} \to -\infty$ - $3r^{-0.1} = 3 \times \frac{1}{r^{0.1}} \to +\infty$ - $-4r^{-\frac{3}{7}} = -4 \times \frac{1}{r^{0.42857}} \to -\infty$ - $-8r^{-0.2} = -8 \times \frac{1}{r^{0.2}} \to -\infty$ 5. **Analyze dominant terms:** The term with the largest negative exponent dominates as $r \to 0$. The exponents are approximately: 0.4, 0.6, 0.1, 0.42857, 0.2. The largest is $0.6$ from $-7r^{-\frac{3}{5}}$. 6. **Conclusion:** Since $-7r^{-\frac{3}{5}} \to -\infty$ dominates, the whole expression tends to $-\infty$. **Final answer:** $$\lim_{r \to 0} \left(5r^{-0.4} - 7r^{-\frac{3}{5}} + 3r^{-0.1} - 4r^{-\frac{3}{7}} - 8r^{-0.2}\right) = -\infty$$