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To solve this limit, we first observe the given expression:

\[
\lim_{x \to 2} \frac{2 - \sqrt{x + 2}}{x^2 - 4}.
\]

### Step-by-step solution:

1. **Substitute \(x = 2\):**
   If we directly substitute \(x = 2\):
   \[
   \frac{2 - \sqrt{2 + 2}}{2^2 - 4} = \frac{2 - \sqrt{4}}{4 - 4} = \frac{2 - 2}{0} = \frac{0}{0},
   \]
   which is an indeterminate form.

2. **Simplify the numerator:**
   Notice that the numerator \(2 - \sqrt{x + 2}\) can be rewritten using a rationalizing technique:
   \[
   2 - \sqrt{x + 2} = \frac{(2 - \sqrt{x + 2})(2 + \sqrt{x + 2})}{2 + \sqrt{x + 2}} = \frac{4 - (x + 2)}{2 + \sqrt{x + 2}}.
   \]
   Simplify \(4 - (x + 2)\):
   \[
   4 - (x + 2) = 2 - x.
   \]
   Therefore, the numerator becomes:
   \[
   2 - \sqrt{x + 2} = \frac{2 - x}{2 + \sqrt{x + 2}}.
   \]

3. **Substitute back into the original limit:**
   Replace the numerator in the original fraction:
   \[
   \lim_{x \to 2} \frac{2 - \sqrt{x + 2}}{x^2 - 4} = \lim_{x \to 2} \frac{\frac{2 - x}{2 + \sqrt{x + 2}}}{x^2 - 4}.
   \]

4. **Simplify the overall fraction:**
   Rewrite the division:
   \[
   \frac{\frac{2 - x}{2 + \sqrt{x + 2}}}{x^2 - 4} = \frac{2 - x}{(2 + \sqrt{x + 2})(x^2 - 4)}.
   \]
   Notice that \(x^2 - 4 = (x - 2)(x + 2)\), so:
   \[
   \frac{2 - x}{(2 + \sqrt{x + 2})(x - 2)(x + 2)}.
   \]

5. **Simplify further:**
   Since \(2 - x = -(x - 2)\), the expression simplifies to:
   \[
   \frac{-(x - 2)}{(2 + \sqrt{x + 2})(x - 2)(x + 2)} = \frac{-1}{(2 + \sqrt{x + 2})(x + 2)}.
   \]

6. **Take the limit as \(x \to 2\):**
   Substitute \(x = 2\) into the simplified expression:
   \[
   \frac{-1}{(2 + \sqrt{2 + 2})(2 + 2)} = \frac{-1}{(2 + 2)(4)} = \frac{-1}{16}.
   \]

### Final Answer:
\[
\boxed{-0.0625}
\]

Would you like further clarification or a graphical representation? Let me know! 

### Follow-up questions:
1. What happens if we solve this problem using numerical approaches?
2. Can this limit be solved without rationalization?
3. How does substituting values close to \(x = 2\) help in verifying this result?
4. Can we rewrite the limit using L'Hopital's rule?
5. What are other common indeterminate forms, and how are they handled?

### Tip:
Always check for indeterminate forms (\(\frac{0}{0}\), \(\infty - \infty\), etc.) when solving limits. Rationalizing or factoring are key techniques!

Find the limit of the following: lim_{x→2} (2 - √(x + 2)) / (x^2 - 4). Use decimal numbers only. Keep up to 4 decimal points.

Understand how to solve the limit problem lim_{x→2} (2 - √(x + 2)) / (x^2 - 4) with step-by-step explanations. Learn techniques like rationalization to handle indeterminate forms and find an answer accurate to four decimal places.

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