https://file.aiquickdraw.com/mathbot/file/1727072209639qkyqk9zi.jpg

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!

For the function g whose graph is given, state the value of each quantity, if it exists. (If an answer does not exist, enter DNE.)

Learn how to analyze limits and continuity of a function based on its graph. This problem covers concepts such as left-hand and right-hand limits, conditions for the existence of limits, and how discontinuities affect these values. Suitable for high school students studying Calculus.

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