1. **Problem Statement:** We need to sketch a function $f(x)$ defined on $\mathbb{R}$ that satisfies the following limit conditions: - $\lim_{x \to -4} f(x)$ does not exist (DNE). - $\lim_{x \to -2} f(x) = 0$. - $\lim_{x \to -1} f(x) = 0$. - $\lim_{x \to 0} f(x) = 2$. - $\lim_{x \to 1} f(x)$ DNE. - $\lim_{x \to 2} f(x) = -3$. - $\lim_{x \to 4} f(x) = 5$. - $\lim_{x \to c} f(x) = 5$ for some $c > 4$ with a hole (removable discontinuity) at $x=c$. 2. **Key Concepts and Rules:** - A limit DNE means the function behaves differently from left and right or oscillates near that point. - A limit exists and equals $L$ means the function approaches $L$ from both sides. - A hole (removable discontinuity) means the limit exists but the function is not defined or defined differently at that point. - The function should be continuous everywhere except at points where limit DNE or holes occur. 3. **Constructing the Graph:** - At $x=-4$ and $x=1$, create jump or oscillating discontinuities so limits DNE. - At $x=-2$ and $x=-1$, the function approaches 0 smoothly. - At $x=0$, the function approaches 2. - At $x=2$, the function approaches -3. - At $x=4$, the function approaches 5. - At $x=c > 4$, the function has a hole with limit 5 but no defined value or different value. 4. **Explanation:** - The discontinuities at $x=-4$ and $x=1$ can be jump discontinuities where left and right limits differ. - The points where limits equal 0 or other values can be continuous or removable discontinuities. - The hole at $x=c$ is highlighted to show the limit exists but function value is missing or different. - The function is designed to be continuous elsewhere to satisfy the problem. This sketch satisfies all given limit conditions and discontinuities as required.