1. **State the problem:** We are given a piecewise function $f(x)$ and need to find for each $a = -5, -1, 2, 5$ the values of $f(a)$ and the limit $\lim_{x \to a} f(x)$ if they exist. If a value does not exist, we write "DNE". 2. **Recall definitions:** - $f(a)$ is the value of the function at $x=a$ (the point on the graph). - $\lim_{x \to a} f(x)$ is the value that $f(x)$ approaches as $x$ gets arbitrarily close to $a$ from both sides. 3. **Evaluate each point:** - For $a = -5$: - From the graph, $f(-5)$ is the function value at $x=-5$. The curve is continuous and smooth here, so $f(-5) = 1$. - The limit $\lim_{x \to -5} f(x)$ is also $1$ since the function is continuous at $-5$. - For $a = -1$: - $f(-1)$ is the function value at $x=-1$. The graph shows a point at $y=-6$. - The limit $\lim_{x \to -1} f(x)$ is $-6$ because the function approaches $-6$ from both sides. - For $a = 2$: - $f(2)$ is the function value at $x=2$. The graph shows a point at $y=1$. - The limit $\lim_{x \to 2} f(x)$ is $1$ since the function is continuous and linear near $2$. - For $a = 5$: - $f(5)$ is the function value at $x=5$. The graph shows a point at $y=-2$. - The limit $\lim_{x \to 5} f(x)$ is $-2$ because the function is continuous and linear near $5$. 4. **Summary:** $$ \begin{array}{c|c|c} a & f(a) & \lim_{x \to a} f(x) \\\hline -5 & 1 & 1 \\ -1 & -6 & -6 \\ 2 & 1 & 1 \\ 5 & -2 & -2 \end{array} $$ All values exist and are equal to the function values at those points.