1. **State the problem:** Find the limit $$\lim_{x \to 1} \frac{x + x^2 + x^3 + \cdots + x^{2025} - 2025}{x^2 - x}.$$\n\n2. **Analyze the numerator:** The numerator is a sum of powers of $x$ from $x^1$ to $x^{2025}$ minus 2025. This sum can be written as $$\sum_{k=1}^{2025} x^k - 2025.$$\n\n3. **Recall the sum of a geometric series:** For $x \neq 1$, $$\sum_{k=1}^n x^k = \frac{x(x^n - 1)}{x-1}.$$ Applying this with $n=2025$, we get \n$$\sum_{k=1}^{2025} x^k = \frac{x(x^{2025} - 1)}{x - 1}.$$\n\n4. **Rewrite the original expression:**\n$$\frac{\sum_{k=1}^{2025} x^k - 2025}{x^2 - x} = \frac{\frac{x(x^{2025} - 1)}{x - 1} - 2025}{x^2 - x}.$$\n\n5. **Notice denominator:** $$x^2 - x = x(x-1).$$\n\n6. **Rewrite limit using denominator:**\n$$\lim_{x \to 1} \frac{\frac{x(x^{2025} - 1)}{x - 1} - 2025}{x(x - 1)} = \lim_{x \to 1} \frac{x(x^{2025} - 1) - 2025(x - 1)}{x(x - 1)^2}.$$\n\n7. **Apply substitution:** Let $h = x - 1$, so as $x \to 1$, $h \to 0$ and $x = 1 + h$.\n\n8. **Use binomial expansion for $x^{2025}$ near 1:**\n$$x^{2025} = (1+h)^{2025} \approx 1 + 2025h + \frac{2025 \cdot 2024}{2} h^2 + \cdots.$$\n\n9. **Evaluate numerator near $h=0$:**\n$$x(x^{2025} - 1) = (1 + h)(2025h + \frac{2025\cdot 2024}{2} h^2 + \cdots ) = 2025h + \left(2025 + \frac{2025 \cdot 2024}{2}\right) h^2 + \cdots.$$\n\n10. **Evaluate $2025(x - 1) = 2025h$. So numerator:**\n$$x(x^{2025} - 1) - 2025h = 2025h + \left(2025 + \frac{2025 \cdot 2024}{2}\right) h^2 + \cdots - 2025h = \left(2025 + \frac{2025 \cdot 2024}{2}\right) h^2 + \cdots.$$\n\n11. **So the numerator behaves like a constant times $h^2$ near zero.**\n\n12. **Denominator is:**\n$$x (x - 1)^2 = (1 + h) h^2 = h^2 + h^3.$$\n\n13. **Form the limit:**\n$$\lim_{h \to 0} \frac{\left(2025 + \frac{2025 \cdot 2024}{2}\right) h^2}{h^2 + h^3} = \lim_{h \to 0} \frac{2025 + \frac{2025 \cdot 2024}{2}}{1 + h} = 2025 + \frac{2025 \cdot 2024}{2}.$$\n\n14. **Calculate final value:**\n$$2025 + \frac{2025 \times 2024}{2} = 2025 + 2025 \times 1012 = 2025 (1 + 1012) = 2025 \times 1013.$$\nCalculate $2025 \times 1013$:\n$2025 \times 1000 = 2,025,000$,\n$2025 \times 13 = 26,325$,\nSum = $2,025,000 + 26,325 = 2,051,325.$\n\n**Final answer:** $$\boxed{2051325}.$$