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The problem involves determining limits by using a table of values, followed by an analysis based on those limits. Let's break down the problem and guide the completion process:

### Function for Limit Analysis:
\[
\lim_{x \to 2} \frac{1 - \frac{2}{x}}{x^2 - 4}
\]

### Instructions for the Table:
The table includes values of \( x \) approaching 2 from both sides, as well as the corresponding values of \( \frac{1 - \frac{2}{x}}{x^2 - 4} \). 

The given task is broken into three parts:

1. **Part (a)**: Use the table to determine \( \lim_{x \to 2^-} \frac{1 - \frac{2}{x}}{x^2 - 4} \) (approaching from the left).
2. **Part (b)**: Use the table to determine \( \lim_{x \to 2^+} \frac{1 - \frac{2}{x}}{x^2 - 4} \) (approaching from the right).
3. **Part (c)**: Use the results from parts (a) and (b) to determine the overall limit \( \lim_{x \to 2} \frac{1 - \frac{2}{x}}{x^2 - 4} \).

### Next Steps:
1. Let's first complete the table by plugging in the \( x \)-values given.
2. After calculating, we can use the completed table to analyze the left-hand and right-hand limits.

I'll begin by computing the table values for you.Here are the values for the table, based on the function \(\frac{1 - \frac{2}{x}}{x^2 - 4}\):

\[
\begin{array}{|c|c|}
\hline
x & \frac{1 - \frac{2}{x}}{x^2 - 4} \\
\hline
1.9 & 0.13495 \\
1.99 & 0.12594 \\
1.999 & 0.12509 \\
1.9999 & 0.12501 \\
2.1 & 0.11614 \\
2.01 & 0.12407 \\
2.001 & 0.12491 \\
2.0001 & 0.12499 \\
\hline
\end{array}
\]

### Answering the Questions:
- **Part (a)**: As \(x\) approaches 2 from the left (\(x \to 2^-\)), the values in the table are getting closer to approximately **0.125**.
- **Part (b)**: As \(x\) approaches 2 from the right (\(x \to 2^+\)), the values in the table are also approaching **0.125**.
- **Part (c)**: Since the limits from both the left and the right are equal, the overall limit is \( \lim_{x \to 2} \frac{1 - \frac{2}{x}}{x^2 - 4} = 0.125 \).

Would you like further details or clarifications on any step?

Here are some related questions you could explore:
1. What happens if \(x\) approaches 2 for a different function?
2. How does the denominator \(x^2 - 4\) factor into the behavior near \(x = 2\)?
3. What is the significance of one-sided limits in more complex functions?
4. How can the difference between left-hand and right-hand limits affect continuity?
5. Could we use L'Hopital's Rule to confirm the result?

**Tip:** For rational functions that involve a limit approaching a point of discontinuity, it's helpful to evaluate the function numerically, as you've done with this table.

Use the table to determine the limit of (1 - 2/x) / (x^2 - 4) as x approaches 2 from both sides.

Learn how to determine the limit of the function (1 - 2/x) / (x^2 - 4) as x approaches 2. Using a table of values, estimate both the left-hand and right-hand limits. This example is perfect for students learning introductory calculus and limit concepts.

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