Let me analyze the graph to help with the limit and function value calculations based on the given questions.

I'll first examine the graph and then provide the answers to the following:

1. $\lim_{x \to 3} f(x)$
2. $\lim_{x \to 3^+} f(x)$
3. $f(2)$
4. $\lim_{x \to 2^-} f(x)$
5. $\lim_{x \to 2^+} f(x)$

I'll get started by inspecting the graph. PIL import Image

Load the image to analyze the graph

img_path = '/mnt/data/1737687617540t21zsl7t.jpg' img = Image.open(img_path)

Display the image

img.show()I've opened the image with the graph. Now, let's analyze the graph to determine the answers to the questions based on the limits and function values.

I'll start by breaking down each of the questions one by one.

1. $\lim_{x \to 3} f(x)$:

   • The graph shows that as $x$ approaches 3 from both sides (left and right), the function $f(x)$ seems to approach the value of 3. This is because the curve appears to be approaching a value of 3 at $x = 3$ from both sides.

2. $\lim_{x \to 3^+} f(x)$:

   • From the right-hand side of $x = 3$, the function is approaching 3 as well.

3. $f(2)$:

   • At $x = 2$, there is a clear open circle at $y = 2$, indicating that $f(2)$ is not defined. It means the function does not exist at this point.

4. $\lim_{x \to 2^-} f(x)$:

   • From the left of $x = 2$, the function approaches the value of 2.

5. $\lim_{x \to 2^+} f(x)$:

   • From the right of $x = 2$, the function also approaches 2.

Answers:

1. $\lim_{x \to 3} f(x) = 3$
2. $\lim_{x \to 3^+} f(x) = 3$
3. $f(2) = 0.0001$ (since it's not defined, we use 0.0001 as per the instruction)
4. $\lim_{x \to 2^-} f(x) = 2$
5. $\lim_{x \to 2^+} f(x) = 2$

If you have further questions or need a deeper explanation, feel free to ask!

Related questions:

1. What is the significance of an open circle on a graph?
2. How do we define a limit when there is a discontinuity at a point?
3. How can we check if a function is continuous at a point using limits?
4. What is the difference between left-hand and right-hand limits?
5. What do we do when the value of a function is undefined at a point?

Tip: Always look for open circles or jumps on a graph when analyzing limits and function values at specific points!