I see that you've uploaded an image with a graph and several limit and function evaluation questions. Let's address each of the questions based on the graph.

I will first interpret the graph and provide answers to the following:

(a) $\lim_{{x \to 2^-}} f(x)$: This represents the left-hand limit as $x$ approaches 2. We need to find the value of the function as $x$ approaches 2 from the left.

(b) $\lim_{{x \to 2^+}} f(x)$: This represents the right-hand limit as $x$ approaches 2. We need to find the value of the function as $x$ approaches 2 from the right.

(c) $\lim_{{x \to 2}} f(x)$: This is the two-sided limit as $x$ approaches 2. For the limit to exist, the left-hand and right-hand limits must be equal.

(d) $f(2)$: This is the value of the function at $x = 2$.

(e) $\lim_{{x \to 4^-}} f(x)$: This represents the left-hand limit as $x$ approaches 4.

(f) $f(4)$: This is the value of the function at $x = 4$.

I'll now analyze the graph and fill in the answers.

Let me examine the image and provide the answers. PIL import Image

Load the image to examine the graph

image_path = "/mnt/data/17383876539892xkf9jr3.jpg" img = Image.open(image_path) img.show()I have opened the image, and I will now interpret the graph and answer the questions based on it. Let me analyze the limits and function values:

• (a) $\lim_{{x \to 2^-}} f(x)$: The left-hand limit at $x = 2$ is the value the function approaches from the left side. In the graph, it appears that as $x$ approaches 2 from the left, the function is approaching 2.

• (b) $\lim_{{x \to 2^+}} f(x)$: The right-hand limit at $x = 2$ is the value the function approaches from the right side. The graph shows that as $x$ approaches 2 from the right, the function approaches a value near 3.

• (c) $\lim_{{x \to 2}} f(x)$: The two-sided limit at $x = 2$ only exists if the left and right limits are equal. Since the left-hand limit is 2 and the right-hand limit is 3, this two-sided limit does not exist.

• (d) $f(2)$: The value of the function at $x = 2$ is given by the open circle at $x = 2$, which is 2 (since it's an open circle, it does not include 2, but the value approaches it).

• (e) $\lim_{{x \to 4^-}} f(x)$: The left-hand limit at $x = 4$ appears to be approaching a value of 3 as $x$ approaches 4 from the left.

• (f) $f(4)$: The value of the function at $x = 4$ is the closed circle at $x = 4$, which is 4.

So, here are the answers:

• (a) $\lim_{{x \to 2^-}} f(x) = 2$
• (b) $\lim_{{x \to 2^+}} f(x) = 3$
• (c) $\lim_{{x \to 2}} f(x) = \text{DNE}$ (Does Not Exist)
• (d) $f(2) = 2$
• (e) $\lim_{{x \to 4^-}} f(x) = 3$
• (f) $f(4) = 4$

Would you like more details or further clarification on any of these answers?