1. **State the problem:** We are asked to find various limits and function values of the function $h(x)$ at points $x = -3$, $x = 0$, and as $x \to \infty$.\n\n2. **Recall limit definitions:**\n- The left-hand limit $\lim_{x \to a^-} h(x)$ is the value $h(x)$ approaches as $x$ approaches $a$ from the left.\n- The right-hand limit $\lim_{x \to a^+} h(x)$ is the value $h(x)$ approaches as $x$ approaches $a$ from the right.\n- The limit $\lim_{x \to a} h(x)$ exists if and only if the left and right limits are equal.\n- The function value $h(a)$ is the value of the function at $x=a$.\n\n3. **Evaluate each part using the graph description:**\n\n(a) $\lim_{x \to -3^-} h(x)$: Approaching $-3$ from the left, the graph rises steeply to an open circle at about $y=7$. So, $\lim_{x \to -3^-} h(x) = 7$.\n\n(b) $\lim_{x \to -3^+} h(x)$: Approaching $-3$ from the right, the graph dips sharply below the x-axis, so the limit is less than 0. The exact value is not given, but it is clearly below zero. We denote it as some value $L < 0$.\n\n(c) $\lim_{x \to -3} h(x)$: Since left and right limits differ ($7$ vs $L<0$), the limit does not exist.\n\n(d) $h(-3)$: The graph shows an open circle at $x=-3$, so $h(-3)$ is undefined or not equal to the limit values.\n\n(e) $\lim_{x \to 0^-} h(x)$: Approaching $0$ from the left, the graph crosses the x-axis near zero with a closed point at zero, so the limit from the left is $0$.\n\n(f) $\lim_{x \to 0^+} h(x)$: Approaching $0$ from the right, the graph also crosses the x-axis at the closed point, so the limit from the right is $0$.\n\n(g) $\lim_{x \to \infty} h(x)$: As $x$ goes to infinity, the graph oscillates rapidly between $x=2$ and $x=5$, then rises to a peak near $x=6-7$ and falls slightly near $x=8$. Since the graph does not settle to a single value, the limit does not exist.\n\n(h) $h(0)$: The graph has a closed point at zero on the x-axis, so $h(0) = 0$.\n\n(i) $\lim_{x \to \infty} h(x)$: Same as (g), the limit does not exist due to oscillations and no settling value.\n\n**Final answers:**\n(a) $7$\n(b) Does not equal 7, less than 0 (exact value unknown)\n(c) Does not exist\n(d) Undefined (open circle)\n(e) $0$\n(f) $0$\n(g) Does not exist\n(h) $0$\n(i) Does not exist