Step 1: Divide every term in the numerator and the denominator by the highest power of $K$, which is $K^2$. $$ \lim_{K \to \infty} \frac{K^2 - 2K + 3}{2K^2 + 5K - 3} = \lim_{K \to \infty} \frac{\frac{K^2}{K^2} - \frac{2K}{K^2} + \frac{3}{K^2}}{\frac{2K^2}{K^2} + \frac{5K}{K^2} - \frac{3}{K^2}} $$ Step 2: Simplify the terms. $$ \lim_{K \to \infty} \frac{1 - \frac{2}{K} + \frac{3}{K^2}}{2 + \frac{5}{K} - \frac{3}{K^2}} $$ Step 3: Apply the limit. As $K \to \infty$, terms of the form $\frac{c}{K^n}$ (where $n > 0$) approach $0$. $$ \frac{1 - 0 + 0}{2 + 0 - 0} $$ Step 4: Calculate the final value. $$ \frac{1}{2} $$ The limit is: $$ \boxed{\frac{1}{2}} $$