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:

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.