1. **Problem Statement:** Evaluate the left-hand limit, right-hand limit, and two-sided limit of the function $f(x)$ at $x = -3, -1, 1, 4$ using the graph. Also, find the function values $f(-3), f(-1), f(1), f(4)$. 2. **Recall:** The left-hand limit $\lim_{x \to a^-} f(x)$ is the value $f(x)$ approaches as $x$ approaches $a$ from the left. The right-hand limit $\lim_{x \to a^+} f(x)$ is the value $f(x)$ approaches as $x$ approaches $a$ from the right. The two-sided limit $\lim_{x \to a} f(x)$ exists only if both one-sided limits exist and are equal. 3. **Evaluate at $x = -3$:** - From the graph, as $x \to -3^-$, $f(x)$ approaches the $y$-value of the left rising line segment ending in an open circle near $-3$. This value is approximately $2$. - As $x \to -3^+$, $f(x)$ approaches the start of the left-middle curved piece from an open circle near $-3$ down to a filled point near $-1$, which is about $-1$. - Since $\lim_{x \to -3^-} f(x) \neq \lim_{x \to -3^+} f(x)$, the two-sided limit $\lim_{x \to -3} f(x) = \text{DNE}$. 4. **Evaluate at $x = -1$:** - As $x \to -1^-$, $f(x)$ approaches the filled point near $-1$, which is about $-2$. - As $x \to -1^+$, $f(x)$ approaches the open circle near $-1$ on the middle smooth curve, which is about $0$. - Since the one-sided limits differ, $\lim_{x \to -1} f(x) = \text{DNE}$. 5. **Evaluate at $x = 1$:** - As $x \to 1^-$, $f(x)$ approaches the open circle near $1$ on the middle smooth curve crossing the $x$-axis, so the limit is $0$. - As $x \to 1^+$, $f(x)$ also approaches the same open circle near $1$, so the limit is $0$. - Since both one-sided limits are equal, $\lim_{x \to 1} f(x) = 0$. 6. **Evaluate at $x = 4$:** - As $x \to 4^-$, $f(x)$ approaches the open circle near $4$ on the middle smooth curve, approximately $1$. - As $x \to 4^+$, $f(x)$ approaches the filled point below the $x$-axis at $4$, approximately $-1$. - Since the one-sided limits differ, $\lim_{x \to 4} f(x) = \text{DNE}$. 7. **Function values:** - $f(-3)$ is the value at $x = -3$. The graph shows an open circle at $-3$ on the left rising line segment (value $2$) and a filled point on the left-middle curve near $-1$. Since the filled point is not at $-3$, $f(-3) = \text{Undefined}$. - $f(-1)$ is the filled point near $-1$, which is $-2$. - $f(1)$ is the value at $x=1$. The graph shows an open circle at $1$ on the middle smooth curve, so $f(1) = \text{Undefined}$. - $f(4)$ is the filled point below the $x$-axis at $4$, approximately $-1$. **Final answers:** (a) $\lim_{x \to -3^-} f(x) = 2$, $\lim_{x \to -3^+} f(x) = -1$, $\lim_{x \to -3} f(x) = \text{DNE}$ (b) $\lim_{x \to -1^-} f(x) = -2$, $\lim_{x \to -1^+} f(x) = 0$, $\lim_{x \to -1} f(x) = \text{DNE}$ (c) $\lim_{x \to 1^-} f(x) = 0$, $\lim_{x \to 1^+} f(x) = 0$, $\lim_{x \to 1} f(x) = 0$ (d) $\lim_{x \to 4^-} f(x) = 1$, $\lim_{x \to 4^+} f(x) = -1$, $\lim_{x \to 4} f(x) = \text{DNE}$ (e) $f(-3) = \text{Undefined}$, $f(-1) = -2$, $f(1) = \text{Undefined}$, $f(4) = -1$