1. **Problem:** Calculate the limit $$\lim_{x \to -1} \frac{\sqrt{x^3 - 2}}{x - 1}$$. 2. **Formula and rules:** To find limits involving square roots and rational expressions, check if direct substitution leads to an indeterminate form. If so, consider algebraic manipulation such as rationalizing the numerator or denominator. 3. **Step 1: Direct substitution** Substitute $x = -1$: $$\sqrt{(-1)^3 - 2} = \sqrt{-1 - 2} = \sqrt{-3}$$ which is not a real number. Since the expression under the root is negative, the function is not defined for $x = -1$ in real numbers. 4. **Step 2: Domain consideration** The expression inside the square root is $x^3 - 2$. For real values, $x^3 - 2 \geq 0 \Rightarrow x^3 \geq 2 \Rightarrow x \geq \sqrt[3]{2} \approx 1.26$. Since $-1 < 1.26$, the function is not defined near $x = -1$ in the real domain. 5. **Conclusion:** The limit $$\lim_{x \to -1} \frac{\sqrt{x^3 - 2}}{x - 1}$$ does not exist in the real numbers because the function is not defined near $x = -1$. Final answer: The limit does not exist (DNE) in real numbers.