To estimate $\lim_{x \to 4} g(x)$ based on the table of values:

• As $x$ approaches 4 from the left (3.9, 3.99, 3.999), the values of $g(x)$ approach 12.01.
• As $x$ approaches 4 from the right (4.001, 4.01, 4.1), the values of $g(x)$ also approach 12.01.

Since the values of $g(x)$ around $x = 4$ are close to 12.01 from both sides, a reasonable estimate for $\lim_{x \to 4} g(x)$ is:

$\lim_{x \to 4} g(x) \approx 12.01$

Would you like a detailed explanation of how limits work or any clarifications on the process?

Here are 5 related questions to deepen understanding:

1. How do we define a limit mathematically?
2. What is the difference between left-hand and right-hand limits?
3. How would the behavior of $g(x)$ change if there was a discontinuity at $x = 4$?
4. How can we verify this estimate graphically or analytically?
5. What are some other methods for estimating limits besides using tables of values?

Tip: Always check for both left-hand and right-hand limits when estimating limits from data, to ensure they converge to the same value.