1. **State the problem:** Find the limit $$\lim_{x \to 3} \frac{\sqrt{x^3 - x - 1} - \sqrt{x^3 - 7x + 7}}{\sqrt{x^3 - x - 1} - \sqrt{x^3 - 3x + 3}}.$$\n\n2. **Identify the form:** Substitute $x=3$:\nCalculate each expression under the roots:\n$$3^3 - 3 - 1 = 27 - 3 - 1 = 23,$$\n$$3^3 - 7\cdot3 + 7 = 27 - 21 + 7 = 13,$$\n$$3^3 - 3\cdot3 + 3 = 27 - 9 + 3 = 21.$$\nSo the numerator is $\sqrt{23} - \sqrt{13}$ and the denominator is $\sqrt{23} - \sqrt{21}$. Since these are not zero, the limit is a direct substitution:\n\n3. **Evaluate the limit by direct substitution:**\n$$\lim_{x \to 3} \frac{\sqrt{23} - \sqrt{13}}{\sqrt{23} - \sqrt{21}} = \frac{\sqrt{23} - \sqrt{13}}{\sqrt{23} - \sqrt{21}}.$$\n\n4. **Simplify the expression if desired:** This is the exact value of the limit.\n\n**Final answer:** $$\boxed{\frac{\sqrt{23} - \sqrt{13}}{\sqrt{23} - \sqrt{21}}}.$$