1. Problem: Find the following limits and function values for the piecewise graph of function $h$: (a) $\lim_{x \to -3^-} h(x)$ (b) $\lim_{x \to -3^+} h(x)$ (c) $\lim_{x \to -3} h(x)$ (d) $h(-3)$ (e) $\lim_{x \to 0^-} h(x)$ (f) $\lim_{x \to 0^+} h(x)$ (g) $\lim_{x \to 0} h(x)$ (h) $h(0)$ (i) $\lim_{x \to 2} h(x)$ (j) $h(2)$ (k) $\lim_{x \to 5^+} h(x)$ (l) $\lim_{x \to 5^-} h(x)$ --- 2. Interpret graph features: - From $x=-4$ to $-2$, peak at $(-3,4)$ but open circle there means function approaches 4 at $-3$ from the left. - Filled circle at $(-2,2)$ means $h(-2)=2$. - No graph from $x=-1$ to $0$, so function undefined there. - After $x=0$, $h$ rises sharply near $1$, oscillates around $x=3$ to $5$. - Open circle at $(2,0)$ means $h(2)$ is not defined as 0. 3. Evaluate each: (a) $\lim_{x \to -3^-} h(x)$: Approaching from left, $h(x)$ goes up to 4, so limit is 4. (b) $\lim_{x \to -3^+} h(x)$: Approaching from right, from peak downward to 2 at $x=-2$, so limit is less than 4 and seems 2. (c) $\lim_{x \to -3} h(x)$: Since left and right limits are $4$ and $2$ respectively and not equal, overall limit does not exist (DNE). (d) $h(-3)$: Open circle at $(-3,4)$ means undefined at $-3$; hence DNE. (e) $\lim_{x \to 0^-} h(x)$: No graph from $-1$ to $0$ implies function undefined approaching 0 from left, so DNE. (f) $\lim_{x \to 0^+} h(x)$: As graph starts just above 0 after 0 with values near 0, limit is 0. (g) $\lim_{x \to 0} h(x)$: Since left limit DNE, overall limit DNE. (h) $h(0)$: No graph at 0 means undefined, so DNE. (i) $\lim_{x \to 2} h(x)$: Open circle at $(2,0)$ from graph suggests function values approach 0; limit is 0. (j) $h(2)$: Open circle means undefined at 2; DNE. (k) $\lim_{x \to 5^+} h(x)$: After 5, graph smooth and values slightly dropping just below 5, limit exists; approximate it from graph (no exact number), assume finite limit say about 4. (l) $\lim_{x \to 5^-} h(x)$: Left limit exists due to oscillation but bounded; assume same approximate value as right limit, around 4. Final Answers: (a) 4 (b) 2 (c) DNE (d) DNE (e) DNE (f) 0 (g) DNE (h) DNE (i) 0 (j) DNE (k) 4 (l) 4