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The problem asks to evaluate the following limit as \( x \to \infty \):

\[
\lim_{x \to \infty} \frac{3x^2 + 2x}{\sqrt{x^2 - 2x}} 
\]

We can simplify and analyze this expression step by step:

### Step 1: Factor the dominant terms
Both the numerator and denominator involve terms that grow quadratically as \( x \to \infty \). To handle this, factor out \( x^2 \) from both the numerator and the square root in the denominator.

#### Numerator:
\[
3x^2 + 2x \approx x^2(3 + \frac{2}{x})
\]

#### Denominator:
\[
\sqrt{x^2 - 2x} = \sqrt{x^2(1 - \frac{2}{x})} = x\sqrt{1 - \frac{2}{x}}
\]

### Step 2: Simplify the expression
Now we can rewrite the entire limit expression:

\[
\lim_{x \to \infty} \frac{x^2 \left( 3 + \frac{2}{x} \right)}{x \sqrt{1 - \frac{2}{x}}}
\]

The \( x^2 \) in the numerator and the \( x \) in the denominator allow us to simplify this to:

\[
\lim_{x \to \infty} \frac{x \left( 3 + \frac{2}{x} \right)}{\sqrt{1 - \frac{2}{x}}}
\]

### Step 3: Evaluate the limit as \( x \to \infty \)
As \( x \to \infty \), the term \( \frac{2}{x} \) in both the numerator and denominator approaches 0. So, the expression becomes:

\[
\lim_{x \to \infty} \frac{x(3 + 0)}{\sqrt{1 - 0}} = \lim_{x \to \infty} \frac{3x}{1} = 3x
\]

Thus, the limit tends to infinity as \( x \to \infty \).

### Interpretation of the result:
Since the limit tends to infinity, it suggests that the function does not have a horizontal asymptote as \( x \to \infty \). Instead, the function grows without bound, indicating that the function has no horizontal asymptotic behavior in the positive infinity direction.

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Do you want more details or have any questions about this solution?

Here are some related questions for further practice:

1. How would the limit change if the denominator were \( \sqrt{x^2 + 2x} \)?
2. What happens if we consider \( \lim_{x \to -\infty} \frac{3x^2 + 2x}{\sqrt{x^2 - 2x}} \)?
3. Can you explain how to handle indeterminate forms in limits at infinity?
4. What does it mean for a function to have a slant asymptote instead of a horizontal one?
5. How does factoring the highest power of \( x \) help simplify limits at infinity?

**Tip:** When evaluating limits at infinity, always factor out the highest power of \( x \) from both the numerator and denominator to simplify the expression.

Find the limit as x approaches infinity of (3x^2 + 2x) / (sqrt(x^2 - 2x)). What does this result say about the existence of horizontally asymptotic behavior as x tends to infinity? Show your work and justify your reasoning.

Learn how to evaluate the limit of (3x^2 + 2x) / (sqrt(x^2 - 2x)) as x tends to infinity. This detailed step-by-step solution covers key calculus concepts including limits at infinity and asymptotic behavior. Suitable for high school students in Grades 11-12.

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