Looking at the graph of the function $g(t)$, I will now fill in the blanks for each limit or value:

(a) $\lim_{t \to 0^-} g(t)$: As $t$ approaches 0 from the left, the function approaches $-1$. So, the answer is: $\boxed{-1}$

(b) $\lim_{t \to 0^+} g(t)$: As $t$ approaches 0 from the right, the function approaches $-2$. So, the answer is: $\boxed{-2}$

(c) $\lim_{t \to 0} g(t)$: Since the left-hand limit $(-1)$ and right-hand limit $(-2)$ are not equal, the limit does not exist (DNE). So, the answer is: $\boxed{\text{DNE}}$

(d) $\lim_{t \to 2^-} g(t)$: As $t$ approaches 2 from the left, the function approaches $2$. So, the answer is: $\boxed{2}$

(e) $\lim_{t \to 2^+} g(t)$: As $t$ approaches 2 from the right, the function approaches $0$. So, the answer is: $\boxed{0}$

(f) $\lim_{t \to 2} g(t)$: Since the left-hand limit $(2)$ and right-hand limit $(0)$ are not equal, the limit does not exist (DNE). So, the answer is: $\boxed{\text{DNE}}$

(g) $g(2)$: There is a hole in the graph at $t = 2$, so the function is not defined at this point. So, the answer is: $\boxed{\text{DNE}}$

(h) $\lim_{t \to 4} g(t)$: As $t$ approaches 4 from both sides, the function approaches $3$. So, the answer is: $\boxed{3}$

Let me know if you'd like a detailed explanation for any of these limits or if you have questions!

Here are some related questions for further practice:

1. What is the significance of left-hand and right-hand limits?
2. Why does a limit not exist when the left-hand and right-hand limits are different?
3. How do discontinuities like holes and jumps affect the existence of limits?
4. Can a function be continuous at a point if the limit exists but the function is not defined at that point?
5. How can you determine the value of a function at a specific point from its graph?

Tip: For a function to have a limit at a point, the left-hand and right-hand limits must both exist and be equal!