Let's analyze the graph you've uploaded to answer the questions.

(A) Find $\lim_{x \to 3^-} f(x)$

This asks for the left-hand limit of the function as $x$ approaches 3. From the graph, we need to look at the behavior of the function as $x$ gets closer to 3 from values smaller than 3.

• The graph shows that as $x \to 3^-$, the function approaches 2 (based on the horizontal level the function is approaching).

So, the left-hand limit is: $\lim_{x \to 3^-} f(x) = 2$

(B) Find $\lim_{x \to 3^+} f(x)$

This asks for the right-hand limit of the function as $x$ approaches 3. From the graph, we need to look at the behavior of the function as $x$ gets closer to 3 from values greater than 3.

• The graph shows that as $x \to 3^+$, the function approaches negative infinity.

So, the right-hand limit is: $\lim_{x \to 3^+} f(x) = -\infty$

(C) Find $\lim_{x \to 3} f(x)$

This asks for the two-sided limit of the function as $x$ approaches 3. To determine this, both the left-hand and right-hand limits need to exist and be equal.

• From part (A), the left-hand limit is 2.
• From part (B), the right-hand limit is $-\infty$.

Since these two limits do not match, the two-sided limit does not exist.

So, the two-sided limit is: $\lim_{x \to 3} f(x) = \text{The limit does not exist.}$

(D) Find $f(3)$

This asks for the value of the function at $x = 3$. From the graph, we see that the function is not defined at $x = 3$ (there's a hole at this point).

So, the function is not defined at $x = 3$.

Thus: $f(3) = \text{The function is not defined at } x = 3.$

Summary of Answers:

• (A) $\lim_{x \to 3^-} f(x) = 2$
• (B) $\lim_{x \to 3^+} f(x) = -\infty$
• (C) $\lim_{x \to 3} f(x) = \text{The limit does not exist.}$
• (D) $f(3) = \text{The function is not defined at } x = 3.$

Would you like a more detailed explanation of any of the steps? Here are some additional related questions:

1. How do you find left and right-hand limits for other types of functions?
2. What is the difference between one-sided limits and two-sided limits?
3. Why does the limit not exist when the left and right limits differ?
4. How does a hole in the graph affect the definition of a function at that point?
5. What is the significance of infinity in a limit?

Tip: Always check the behavior of the graph from both sides of the point to determine if the limit exists and if it's finite or infinite.