1. **Problem:** Find the limit $$\lim_{x \to \infty} \frac{x}{2x - 3}$$ 2. **Formula and rules:** When finding limits at infinity for rational functions, divide numerator and denominator by the highest power of $x$ in the denominator to simplify. 3. **Step-by-step solution:** 1. Write the expression: $$\lim_{x \to \infty} \frac{x}{2x - 3}$$ 2. Divide numerator and denominator by $x$ (the highest power of $x$ in the denominator): $$\lim_{x \to \infty} \frac{\cancel{x} \cdot 1}{2 \cancel{x} - 3} = \lim_{x \to \infty} \frac{1}{2 - \frac{3}{x}}$$ 3. As $x \to \infty$, $\frac{3}{x} \to 0$, so: $$\lim_{x \to \infty} \frac{1}{2 - 0} = \frac{1}{2}$$ 4. **Answer:** $$\frac{1}{2}$$ --- 1. **Problem:** Find the limit $$\lim_{x \to \infty} \frac{3x^3 - 5x^2 + 7}{8 + 2x - 5x^3}$$ 2. **Formula and rules:** For limits at infinity of rational functions, divide numerator and denominator by the highest power of $x$ in the denominator. 3. **Step-by-step solution:** 1. Write the expression: $$\lim_{x \to \infty} \frac{3x^3 - 5x^2 + 7}{8 + 2x - 5x^3}$$ 2. Divide numerator and denominator by $x^3$: $$\lim_{x \to \infty} \frac{3 - \frac{5}{x} + \frac{7}{x^3}}{\frac{8}{x^3} + \frac{2}{x^2} - 5}$$ 3. As $x \to \infty$, terms with $\frac{1}{x^n} \to 0$, so: $$\lim_{x \to \infty} \frac{3 - 0 + 0}{0 + 0 - 5} = \frac{3}{-5} = -\frac{3}{5}$$ 4. **Answer:** $$-\frac{3}{5}$$ --- 1. **Problem:** Find the limit $$\lim_{x \to \infty} \frac{x^2 + 3}{x^3 + 2}$$ 2. **Formula and rules:** Divide numerator and denominator by the highest power of $x$ in the denominator. 3. **Step-by-step solution:** 1. Write the expression: $$\lim_{x \to \infty} \frac{x^2 + 3}{x^3 + 2}$$ 2. Divide numerator and denominator by $x^3$: $$\lim_{x \to \infty} \frac{\frac{x^2}{x^3} + \frac{3}{x^3}}{1 + \frac{2}{x^3}} = \lim_{x \to \infty} \frac{\frac{1}{x} + \frac{3}{x^3}}{1 + \frac{2}{x^3}}$$ 3. As $x \to \infty$, $\frac{1}{x} \to 0$ and $\frac{3}{x^3} \to 0$, so: $$\lim_{x \to \infty} \frac{0 + 0}{1 + 0} = 0$$ 4. **Answer:** $$0$$