1. **State the problem:** We are given a function $\varphi(x)$ graphed with behavior near $y=4$ at both ends and vertical asymptotes near $x=0$. We need to find the limits: (a) $\lim_{x \to -\infty} \varphi(x)$ (b) $\lim_{x \to +\infty} \varphi(x)$ 2. **Recall the concept of limits at infinity:** The limit of a function as $x$ approaches $\pm \infty$ describes the behavior of the function far to the left or right on the $x$-axis. 3. **Analyze the graph behavior:** - As $x \to -\infty$, the graph approaches a horizontal line near $y=4$ from slightly below. - As $x \to +\infty$, the graph also approaches $y=4$ from below but more gradually. 4. **Write the limits based on the graph:** $$\lim_{x \to -\infty} \varphi(x) = 4$$ $$\lim_{x \to +\infty} \varphi(x) = 4$$ 5. **Interpretation:** The function $\varphi(x)$ has horizontal asymptotes at $y=4$ on both ends of the $x$-axis. **Final answers:** (a) $4$ (b) $4$