1. **Problem Statement:** Fill in the table with the appropriate values of $\lim_{x \to c} f(x)$ and $f(c)$ based on the given graph. 2. **Understanding Limits and Function Values:** - The limit $\lim_{x \to c} f(x)$ is the value that $f(x)$ approaches as $x$ gets closer to $c$ from both sides. - The function value $f(c)$ is the actual value of the function at $x=c$. - Sometimes, the limit and the function value can be different if there is a jump or removable discontinuity. 3. **Filling the Table:** - For $c=-4$, given: $\lim_{x \to -4} f(x) = 5$, $f(-4) = 3$. - For $c=-2$: From the graph, as $x$ approaches $-2$, $f(x)$ approaches $-3$ from both sides, so $\lim_{x \to -2} f(x) = -3$. The function value $f(-2)$ is at the filled blue dot at $-2$, which is $-3$. - For $c=4$: The limit as $x$ approaches $4$ is $2$ (the curve approaches $2$ from both sides). The function value $f(4)$ is $2$. - For $c=8$: The limit as $x$ approaches $8$ is $1$ (the curve approaches $1$ from both sides). The function value $f(8)$ is $1$. - For $c=10$: The limit as $x$ approaches $10$ is $2$ (the curve approaches $2$). The function value $f(10)$ is $2$. 4. **Summary Table:** | c | $\lim_{x \to c} f(x)$ | $f(c)$ | |----|------------------------|--------| | -4 | 5 | 3 | | -2 | -3 | -3 | | 4 | 2 | 2 | | 8 | 1 | 1 | | 10 | 2 | 2 | 5. **Second Problem: Sketch graphs for given limit and function value conditions at $x=3$:** **a.** $\lim_{x \to 3} f(x)$ does not exist (DNE) and $f(3) = -2$. - This means the left and right limits at $3$ are different or oscillate. - The function value at $3$ is $-2$. **b.** $\lim_{x \to 3} g(x) = 2$ and $g(-3)$ is undefined. - The limit at $3$ is $2$. - The function is not defined at $-3$ (a hole or no point). **c.** $\lim_{x \to 3} h(x)$ does not exist and $h(3) = -2$. - Same as (a), limit does not exist at $3$ but function value is $-2$. 6. **Graph Descriptions:** - For (a) and (c), the graph near $x=3$ has a jump or oscillation causing the limit to not exist, but a filled dot at $(3,-2)$. - For (b), the graph approaches $2$ near $x=3$, and at $x=-3$ there is no point (hole). 7. **Desmos LaTeX for the original function $f(x)$ (approximate):** $$f(x) = \begin{cases} \text{piecewise curve as described, no explicit formula} \end{cases}$$ **For sketches:** - (a) and (c): Use a piecewise function with different left and right limits at $3$ and $f(3)=-2$. - (b): Use a continuous function approaching $2$ at $3$ and undefined at $-3$. Final answers: Table: $c$ | $\lim_{x \to c} f(x)$ | $f(c)$ -4 | 5 | 3 -2 | -3 | -3 4 | 2 | 2 8 | 1 | 1 10 | 2 | 2 Sketches: (a) $\lim_{x \to 3} f(x)$ DNE, $f(3)=-2$ (b) $\lim_{x \to 3} g(x)=2$, $g(-3)$ undefined (c) $\lim_{x \to 3} h(x)$ DNE, $h(3)=-2$