1. **State the problem:** Find the limit $$\lim_{x \to 1} \frac{x^5 - x}{x - 1}$$ using algebraic methods. 2. **Recall the formula and rules:** When direct substitution results in an indeterminate form like $$\frac{0}{0}$$, we use algebraic simplification such as factoring or polynomial division to simplify the expression. 3. **Check direct substitution:** Substitute $x=1$: $$\frac{1^5 - 1}{1 - 1} = \frac{1 - 1}{0} = \frac{0}{0}$$ which is indeterminate. 4. **Factor the numerator:** Notice that $$x^5 - x = x(x^4 - 1)$$. 5. **Factor difference of squares:** $$x^4 - 1 = (x^2)^2 - 1^2 = (x^2 - 1)(x^2 + 1)$$. 6. **Further factor $$x^2 - 1$$:** $$x^2 - 1 = (x - 1)(x + 1)$$. 7. **Rewrite numerator fully factored:** $$x^5 - x = x (x - 1)(x + 1)(x^2 + 1)$$. 8. **Rewrite the original limit expression:** $$\lim_{x \to 1} \frac{x (x - 1)(x + 1)(x^2 + 1)}{x - 1}$$. 9. **Cancel common factor $$x - 1$$:** $$\lim_{x \to 1} \frac{\cancel{(x - 1)} x (x + 1)(x^2 + 1)}{\cancel{(x - 1)}} = \lim_{x \to 1} x (x + 1)(x^2 + 1)$$. 10. **Evaluate the simplified expression at $$x=1$$:** $$1 \times (1 + 1) \times (1^2 + 1) = 1 \times 2 \times 2 = 4$$. **Final answer:** $$\boxed{4}$$