https://file.aiquickdraw.com/mathbot/file/1730580284444yakcq2ki.jpg

Let's analyze the questions in the image one by one.

The image asks for evaluations of specific limits based on the graph of a function \( f(x) \).

### (a) \(\lim_{{x \to 4}} f(x)\)
To evaluate this limit, observe the behavior of \( f(x) \) as \( x \) approaches \( 4 \) from both sides on the graph:
- The limit value appears to be around \( -1 \), since as \( x \) approaches \( 4 \), the \( y \)-values of \( f(x) \) are getting closer to \( -1 \).
  
Thus, the answer is:
\[
\lim_{{x \to 4}} f(x) = -1
\]

### (b) \(\lim_{{x \to -3^+}} f(x)\)
This asks for the right-hand limit of \( f(x) \) as \( x \) approaches \( -3 \) from the positive (right) side.
- Looking at the graph, as \( x \) approaches \( -3 \) from the right, the \( y \)-values approach \( -5 \).

So the answer is:
\[
\lim_{{x \to -3^+}} f(x) = -5
\]

### (c) \(\lim_{{x \to -3^-}} f(x)\)
This asks for the left-hand limit of \( f(x) \) as \( x \) approaches \( -3 \) from the negative (left) side.
- Observing the graph, as \( x \) approaches \( -3 \) from the left, the \( y \)-values appear to approach \( 3 \).

Therefore, the answer is:
\[
\lim_{{x \to -3^-}} f(x) = 3
\]

Would you like further details or explanations on these limit evaluations? 

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### Related Questions:
1. How do we determine if a limit exists at a particular point?
2. What is the difference between a left-hand limit and a right-hand limit?
3. How do we find the limit of a function at a discontinuity?
4. What methods can be used to estimate limits when no graph is provided?
5. Why is the limit notation important in calculus?

#### Tip:
When evaluating limits from a graph, always check both sides of the point to see if they approach the same value. If not, the limit at that point does not exist.

Using the following graph of the function f, evaluate the limits (a) through (c).

Learn how to evaluate the limits of a function using its graph. This problem covers concepts such as left-hand limits, right-hand limits, and determining limit existence based on graphical analysis. Suitable for high school calculus students.

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