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The image displays a table involving a function \( f(x) = \frac{x^2 - 16}{x - 4} \), and the task is to evaluate this function at values of \( x \) near 4, then interpret the results regarding the two-sided limit \( \lim_{x \to 4} f(x) \).

### Let's evaluate the function:
1. The function \( f(x) \) simplifies to:
   \[
   f(x) = \frac{(x-4)(x+4)}{x-4} = x+4, \quad \text{for } x \neq 4
   \]
   So \( f(x) = x + 4 \) for all \( x \neq 4 \).

2. Now, using the simplified version \( f(x) = x + 4 \), let's complete the table:

| \( x \)   | \( f(x) \) |
|-----------|------------|
| 3.9       | \( 3.9 + 4 = 7.9 \) |
| 3.99      | \( 3.99 + 4 = 7.99 \) |
| 3.999     | \( 3.999 + 4 = 7.999 \) |
| 4.1       | \( 4.1 + 4 = 8.1 \) |
| 4.01      | \( 4.01 + 4 = 8.01 \) |
| 4.001     | \( 4.001 + 4 = 8.001 \) |

### Results and Interpretation:
The table shows that as \( x \) approaches 4 from both the left and the right, \( f(x) \) approaches 8. Hence, the two-sided limit is:
\[
\lim_{x \to 4} f(x) = 8
\]

Would you like further details, or do you have any other questions?

### Related Questions:
1. How do we evaluate limits algebraically for functions involving indeterminate forms?
2. What are one-sided limits, and how do they differ from two-sided limits?
3. Why is the function \( f(x) = \frac{x^2 - 16}{x - 4} \) undefined at \( x = 4 \)?
4. How does factorization help in simplifying limits involving polynomials?
5. What are the general strategies for finding limits when direct substitution fails?

### Tip:
When simplifying rational expressions, always check for potential cancellations that can remove indeterminate forms.

Complete the following table for the given function. Round your answers to 4 decimal places, if necessary.

f(x) = (x^2 - 16) / (x - 4)

Table values and limit evaluation based on x approaching 4.

This tutorial helps you evaluate the limit of the function f(x) = (x^2 - 16) / (x - 4) as x approaches 4. Learn how to simplify rational functions and interpret the two-sided limit using a step-by-step solution, suitable for Grade 11-12 students.

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