1. **State the problem:** Find the limit $$\lim_{x \to -r} \frac{\frac{x}{r} - 1}{x + r}$$. 2. **Rewrite the expression:** The numerator is $$\frac{x}{r} - 1 = \frac{x - r}{r}$$. 3. **Substitute into the limit:** $$\lim_{x \to -r} \frac{\frac{x - r}{r}}{x + r} = \lim_{x \to -r} \frac{x - r}{r(x + r)}$$. 4. **Evaluate the limit:** Direct substitution gives denominator $$r(-r + r) = r \cdot 0 = 0$$, so the expression is undefined at $$x = -r$$. 5. **Check for factorization or simplification:** The numerator is $$x - r$$ and denominator is $$r(x + r)$$. No common factors to cancel. 6. **Analyze the behavior near $$x = -r$$:** - Numerator at $$x = -r$$ is $$-r - r = -2r$$. - Denominator approaches zero. 7. **Conclusion:** Since numerator approaches $$-2r$$ (nonzero) and denominator approaches zero, the limit tends to infinity or negative infinity depending on the sign of $$r$$ and the direction of approach. 8. **One-sided limits:** - As $$x \to -r^+$$, $$x + r > 0$$, denominator positive, so limit $$\to -\infty$$ if $$r > 0$$. - As $$x \to -r^-$$, $$x + r < 0$$, denominator negative, so limit $$\to +\infty$$ if $$r > 0$$. **Final answer:** The limit does not exist because the left and right limits are not equal; it diverges to infinity with opposite signs from each side.