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Let's evaluate the limits based on the table provided for the function \( f(x) \). We will analyze how \( f(x) \) behaves as \( x \) approaches -5 from the left (denoted as \( x \to -5^- \)) and from the right (denoted as \( x \to -5^+ \)).

### a) \( \lim_{x \to -5^-} f(x) \)

Looking at the values of \( f(x) \) as \( x \) approaches -5 from the left (i.e., for \( x = -5.1, -5.01, -5.001, -5.0001 \)):
- \( f(-5.1) = 3.9111 \)
- \( f(-5.01) = 3.9911 \)
- \( f(-5.001) = 3.9991 \)
- \( f(-5.0001) = 3.9999 \)

It seems like \( f(x) \) is approaching 4 as \( x \to -5^- \).

Thus, \( \lim_{x \to -5^-} f(x) = 4 \).

### b) \( \lim_{x \to -5^+} f(x) \)

Now, let's examine the values of \( f(x) \) as \( x \) approaches -5 from the right (i.e., for \( x = -4.9999, -4.999, -4.99, -4.9 \)):
- \( f(-4.9999) = -0.4014 \)
- \( f(-4.999) = 0.8331 \)
- \( f(-4.99) = 0.5246 \)
- \( f(-4.9) = -0.7012 \)

Here, the function seems to be fluctuating and not approaching a specific value. Hence, the limit does not exist.

Thus, \( \lim_{x \to -5^+} f(x) = \text{DNE} \) (Does Not Exist).

### c) \( \lim_{x \to -5} f(x) \)

Since the left-hand limit is 4 and the right-hand limit does not exist, the two-sided limit also does not exist.

Thus, \( \lim_{x \to -5} f(x) = \text{DNE} \).

### Summary:
a) \( \lim_{x \to -5^-} f(x) = 4 \)  
b) \( \lim_{x \to -5^+} f(x) = \text{DNE} \)  
c) \( \lim_{x \to -5} f(x) = \text{DNE} \)

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Do you have any questions or would you like further clarification? 

Here are some related questions to explore this topic more deeply:
1. What conditions must be satisfied for a two-sided limit to exist?
2. How can discontinuities in a function affect the existence of limits?
3. What is the difference between left-hand and right-hand limits?
4. Can a function have a one-sided limit without having a two-sided limit? Provide an example.
5. How do you interpret the result when the limit does not exist?

**Tip:** When analyzing limits using tables, always focus on how the function values behave as \( x \) approaches the target value from both directions.

Given the table of values for the function f(x), evaluate the following limits, if they exist. If the limit does not exist, write DNE, ∞, or -∞ as appropriate.

This problem involves evaluating left-hand and right-hand limits for a function f(x) using a table of values. It addresses the existence of limits, continuity, and how to interpret cases where limits do not exist. Suitable for students in Grades 11-12.

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