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The problem is asking for the limit of the function as \( x \) approaches negative infinity:

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

### Solution:

1. **Simplify the Expression:**
   - First, notice that as \( x \) becomes very large in magnitude (negatively), the term \( 2x^2 \) dominates over the constant terms \( 3 \) in both the numerator and the denominator.
   - We can factor \( x^2 \) out of the square root in the numerator.

   \[
   \frac{\sqrt{2x^2 + 3}}{2x + 3} = \frac{\sqrt{x^2(2 + \frac{3}{x^2})}}{2x + 3} = \frac{|x|\sqrt{2 + \frac{3}{x^2}}}{2x + 3}
   \]

2. **Consider the Sign of \( x \):**
   - Since \( x \) is approaching \( -\infty \), \( |x| = -x \). So the expression becomes:

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

3. **Simplify Further:**
   - Divide both the numerator and the denominator by \( x \):

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

4. **Evaluate the Limit:**
   - As \( x \to -\infty \), \( \frac{3}{x^2} \) and \( \frac{3}{x} \) approach 0, simplifying the expression to:

   \[
   \frac{-\sqrt{2}}{2}
   \]

   - Therefore, the limit is:

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

Would you like more details, or do you have any questions about this solution?

### Related Questions:
1. How does the behavior of the function change as \( x \) approaches positive infinity instead of negative infinity?
2. What is the importance of considering the sign of \( x \) when dealing with absolute values in limits?
3. Can you explain why we factor \( x^2 \) out of the square root in the numerator?
4. How would the limit change if the constant in the numerator was a different value?
5. What happens to the limit if the term \( \sqrt{2x^2 + 3} \) was replaced with \( \sqrt{x^4 + 3} \)?

### Tip:
When dealing with limits involving infinity, simplifying the expression by factoring out the highest power of \( x \) can often make the limit more straightforward to evaluate.

Find the limit of the function as x approaches negative infinity: lim(x → -∞) (√(2x² + 3) / (2x + 3)).

Explore the solution to finding the limit as x approaches negative infinity for the function √(2x² + 3) / (2x + 3). The solution involves algebraic simplification and understanding the dominance of leading terms. Suitable for Grades 11-12 or college-level calculus students.

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