1. **State the problem:** Find the limit $$\lim_{x \to 3} \frac{x^2 - 8x + 15}{\sqrt{5x + 34} - 7}$$. 2. **Check direct substitution:** Substitute $x=3$: $$\frac{3^2 - 8(3) + 15}{\sqrt{5(3) + 34} - 7} = \frac{9 - 24 + 15}{\sqrt{15 + 34} - 7} = \frac{0}{\sqrt{49} - 7} = \frac{0}{7 - 7} = \frac{0}{0}$$ which is an indeterminate form. 3. **Use algebraic manipulation:** To resolve the indeterminate form, multiply numerator and denominator by the conjugate of the denominator: $$\frac{x^2 - 8x + 15}{\sqrt{5x + 34} - 7} \times \frac{\sqrt{5x + 34} + 7}{\sqrt{5x + 34} + 7} = \frac{(x^2 - 8x + 15)(\sqrt{5x + 34} + 7)}{(\sqrt{5x + 34} - 7)(\sqrt{5x + 34} + 7)}$$ 4. **Simplify denominator using difference of squares:** $$ (\sqrt{5x + 34})^2 - 7^2 = (5x + 34) - 49 = 5x - 15 $$ 5. **Rewrite the expression:** $$ \frac{(x^2 - 8x + 15)(\sqrt{5x + 34} + 7)}{5x - 15} $$ 6. **Factor numerator and denominator:** $$ x^2 - 8x + 15 = (x - 3)(x - 5) $$ $$ 5x - 15 = 5(x - 3) $$ 7. **Substitute factored forms:** $$ \frac{(x - 3)(x - 5)(\sqrt{5x + 34} + 7)}{5(x - 3)} $$ 8. **Cancel common factor $(x - 3)$:** $$ \frac{\cancel{(x - 3)}(x - 5)(\sqrt{5x + 34} + 7)}{5\cancel{(x - 3)}} = \frac{(x - 5)(\sqrt{5x + 34} + 7)}{5} $$ 9. **Evaluate the limit by substituting $x=3$:** $$ \frac{(3 - 5)(\sqrt{5(3) + 34} + 7)}{5} = \frac{-2(\sqrt{15 + 34} + 7)}{5} = \frac{-2(\sqrt{49} + 7)}{5} = \frac{-2(7 + 7)}{5} = \frac{-2 \times 14}{5} = \frac{-28}{5} $$ **Final answer:** $$ \boxed{-\frac{28}{5}} $$