We are given values of the function $g(x)$ for several values of $x$ that are close to 4 from both the left and the right. Our goal is to estimate $\lim_{x \to 4} g(x)$ based on this data.

Approach:

The function's values near $x = 4$ are:

• As $x$ approaches 4 from the left:

  • $g(3.9) = 11.21$
  • $g(3.99) = 11.92$
  • $g(3.999) = 11.99$

• As $x$ approaches 4 from the right:

  • $g(4.001) = 12.01$
  • $g(4.01) = 12.08$
  • $g(4.1) = 12.81$

Looking at the values near $x = 4$, as $x \to 4^-$, $g(x)$ approaches 11.99, and as $x \to 4^+$, $g(x)$ approaches 12.01. Since these values are very close to each other, it is reasonable to estimate that:

$\lim_{x \to 4} g(x) = 12$

Answer:

The most reasonable estimate for $\displaystyle \lim_{x\to 4}g(x)$ is 12.

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Would you like more details or have any questions? Here are five related questions to expand on this topic:

1. What happens if the function values from the left and right are not close to each other?
2. How would you use this technique if you had fewer data points around $x = 4$?
3. What is the formal definition of a limit and how does it apply to this problem?
4. Can limits exist if the function is discontinuous at a point?
5. How do we estimate limits when dealing with functions that oscillate as $x$ approaches a value?

Tip: When estimating limits from a table of values, check how close the function values are as $x$ approaches the target point from both sides!