1. **Problem Statement:** We are given a function $f(x)$ with vertical asymptotes near $x = -2$ and $x = 2$, and a horizontal asymptote at $y = 5$. We need to find the limits $\lim_{x \to \infty} f(x)$ and $\lim_{x \to -\infty} f(x)$. 2. **Understanding Horizontal Asymptotes:** A horizontal asymptote $y = L$ means that as $x$ approaches $\infty$ or $-\infty$, the function $f(x)$ approaches the value $L$. Formally, $$\lim_{x \to \infty} f(x) = L \quad \text{or} \quad \lim_{x \to -\infty} f(x) = L.$$ 3. **Given Information:** The horizontal asymptote is $y = 5$, and the graph shows the curve approaching $y=5$ from above as $x$ goes to both positive and negative infinity. 4. **Conclusion:** Therefore, $$\lim_{x \to \infty} f(x) = 5$$ $$\lim_{x \to -\infty} f(x) = 5$$ 5. **Summary:** The function $f(x)$ approaches the horizontal asymptote $y=5$ as $x$ tends to both positive and negative infinity.