1. The problem is to find the limit $$\lim_{x \to -\infty} \frac{5x + 1}{2x}$$. 2. The formula for limits involving rational functions as $x$ approaches infinity or negative infinity is to divide numerator and denominator by the highest power of $x$ in the denominator. 3. Divide numerator and denominator by $x$: $$\frac{5x + 1}{2x} = \frac{\frac{5x}{x} + \frac{1}{x}}{\frac{2x}{x}} = \frac{5 + \frac{1}{x}}{2}$$ 4. As $x \to -\infty$, $\frac{1}{x} \to 0$, so the expression simplifies to: $$\frac{5 + 0}{2} = \frac{5}{2}$$ 5. Therefore, the limit is: $$\lim_{x \to -\infty} \frac{5x + 1}{2x} = \frac{5}{2}$$ This means as $x$ becomes very large in the negative direction, the value of the function approaches $2.5$.