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!