Let's analyze the graph in the image and determine the required values for the limits and the function at specific points. Based on the questions, we are asked to evaluate limits and function values of $g(t)$ at various points.

Function $g(t)$ Observations:

From the graph:

• The curve $g(t)$ crosses the x-axis at several points and has a smooth behavior except at certain key points.
• We have to evaluate limits as $t$ approaches specific values from both the left and the right, and also directly calculate function values where possible.

Questions:

• (a) $\lim_{t \to 0} g(t)$:

  • As $t$ approaches 0, $g(t)$ seems to approach a value of 0 from both sides. Therefore: $\lim_{t \to 0} g(t) = 0$

• (b) $\lim_{t \to 3^+} g(t)$:

  • As $t$ approaches 3 from the right, $g(t)$ increases sharply and seems to approach a value close to 4. Thus: $\lim_{t \to 3^+} g(t) = 4$

• (c) $\lim_{t \to 3^-} g(t)$:

  • As $t$ approaches 3 from the left, $g(t)$ is decreasing and seems to approach around 2. Therefore: $\lim_{t \to 3^-} g(t) = 2$

• (d) $\lim_{t \to 2} g(t)$:

  • At $t = 2$, there is a clear point on the graph where the function exists, and it looks like $g(2) = 1.5$ (between 1 and 2). So: $\lim_{t \to 2} g(t) = 1.5$

• (e) $\lim_{t \to -1^-} g(t)$:

  • As $t$ approaches $-1$ from the left, the function decreases and appears to approach a value around -2. Therefore: $\lim_{t \to -1^-} g(t) = -2$

• (f) $\lim_{t \to 2^-} g(t)$:

  • As $t$ approaches 2 from the left, $g(t)$ also appears to be approximately 1.5. So: $\lim_{t \to 2^-} g(t) = 1.5$

• (g) $g(2)$:

  • The function value at $t = 2$ is directly seen to be around 1.5, as the graph passes through that point.

If you need further clarifications or a more detailed explanation, feel free to ask!

Related Questions:

1. How do you evaluate one-sided limits from a graph?
2. What is the significance of the value of $g(t)$ at $t = 2$?
3. Can a function have different one-sided limits at a given point?
4. How do we know if a limit does not exist from a graph?
5. What is the importance of continuous and discontinuous points on the graph?

Tip:

When evaluating limits from a graph, always look for sharp jumps, asymptotes, or breaks that indicate discontinuities or differing behavior from the left and right sides.