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We are tasked with solving the following limit:

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

Let's work through the steps:

### Step 1: Analyze the degree of terms
The highest degree term in both the numerator and the denominator is \( x^2 \), so we can factor \( x \) out of both to simplify.

### Step 2: Factor \( x \) from the numerator and denominator
In the numerator, \( x - 2 \), we factor out \( x \):
\[
x - 2 = x \left(1 - \frac{2}{x}\right)
\]

In the denominator, we have \( \sqrt{x^2 + x - 2} \). We factor \( x^2 \) from the expression inside the square root:
\[
\sqrt{x^2 + x - 2} = \sqrt{x^2 \left(1 + \frac{1}{x} - \frac{2}{x^2}\right)} = x \sqrt{1 + \frac{1}{x} - \frac{2}{x^2}}
\]

### Step 3: Simplify the limit expression
Now, rewrite the limit as:
\[
\lim_{x \to \infty} \frac{x \left(1 - \frac{2}{x}\right)}{x \sqrt{1 + \frac{1}{x} - \frac{2}{x^2}}}
\]

Cancel out \( x \) from both the numerator and denominator:
\[
\lim_{x \to \infty} \frac{1 - \frac{2}{x}}{\sqrt{1 + \frac{1}{x} - \frac{2}{x^2}}}
\]

### Step 4: Take the limit as \( x \to \infty \)
As \( x \to \infty \), \( \frac{2}{x} \to 0 \), \( \frac{1}{x} \to 0 \), and \( \frac{2}{x^2} \to 0 \). Thus, the expression simplifies to:
\[
\frac{1 - 0}{\sqrt{1 + 0 - 0}} = \frac{1}{\sqrt{1}} = 1
\]

### Conclusion
The limit evaluates to **1**. Therefore, the correct answer is:

\[
\boxed{D. \ 1}
\]

Would you like further details on any of the steps, or do you have any questions?

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Here are 5 related questions to extend this topic:

1. How would the limit change if the term inside the square root had a different coefficient for \(x^2\)?
2. What if the expression involved \( x \to -\infty \) instead of \( x \to \infty \)?
3. How do we handle limits at infinity for rational functions with different degrees in the numerator and denominator?
4. How can L'Hopital's Rule be applied to similar limits?
5. What are the general strategies for simplifying limits involving square roots?

**Tip**: Whenever dealing with limits at infinity, especially with square roots, try factoring out the dominant term to simplify the expression.

lim_{x \to \infty} \frac{x - 2}{\sqrt{x^2 + x - 2}}

Learn how to solve the limit of (x - 2) over sqrt(x^2 + x - 2) as x approaches infinity, with detailed steps focusing on factoring and simplification of square roots. Key algebraic and limit-based concepts are covered.

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