1. **State the problem:** Find the limit as $x \to \infty$ of the expression $$\frac{x^2 + 4x}{2x^2 - x}.$$\n\n2. **Rewrite the expression:** To simplify, divide numerator and denominator by $x^2$, the highest power of $x$ in the denominator:\n$$\frac{\frac{x^2}{x^2} + \frac{4x}{x^2}}{\frac{2x^2}{x^2} - \frac{x}{x^2}} = \frac{1 + \frac{4}{x}}{2 - \frac{1}{x}}.$$\n\n3. **Analyze the limit:** As $x \to \infty$, the terms $\frac{4}{x} \to 0$ and $\frac{1}{x} \to 0$.\n\n4. **Evaluate the limit:** Substitute these limits into the simplified expression:\n$$\lim_{x \to \infty} \frac{1 + \frac{4}{x}}{2 - \frac{1}{x}} = \frac{1 + 0}{2 - 0} = \frac{1}{2}.$$\n\n**Final answer:** $$\boxed{\frac{1}{2}}.$$