Let's analyze and solve the limits based on the provided graph for each part (a) through (c):

1. Part (a): $\lim_{{x \to 4}} f(x)$

   • Observing the graph at $x = 4$, we should look at the behavior of $f(x)$ as $x$ approaches 4 from both the left and the right.
   • The values appear to approach a specific point from both sides. Based on the options provided, it seems that the correct choice should be identified based on what we see on the graph.

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

   • This denotes the right-hand limit as $x$ approaches -3 from the positive (right) side.
   • Observing the function's behavior from the right of $x = -3$, we determine whether the function value approaches -5 or another value.

3. Part (c): $\lim_{{x \to -3^-}} f(x)$

   • This denotes the left-hand limit as $x$ approaches -3 from the negative (left) side.
   • Observing the function's behavior from the left of $x = -3$, we determine whether there is a specific value for the limit.

Solutions

Let’s go through each part based on these observations from the graph:

• (a) Select the correct choice for $\lim_{{x \to 4}} f(x)$ and input the answer within the choice.
• (b) Write the value of $\lim_{{x \to -3^+}} f(x)$.
• (c) Write the value of $\lim_{{x \to -3^-}} f(x)$.

Would you like further guidance on interpreting the graph for these values? Let me know if you'd like a detailed explanation of each limit.

Here are some related questions to deepen your understanding:

1. What is the difference between right-hand and left-hand limits?
2. How do you determine if a two-sided limit exists based on a graph?
3. What does it mean when a function has a jump discontinuity at a point?
4. How can you determine if a function is continuous at a specific point?
5. Why do limits approaching from different directions sometimes have different values?

Tip: Always check the graph's behavior on both sides of the point for a limit. A two-sided limit only exists if both one-sided limits are equal.