To evaluate the limit 

\[
\lim_{x \to \infty} \frac{\sqrt{8x^2 + 7}}{3 - 2x},
\]

we can analyze the behavior of the numerator and the denominator as \(x\) approaches infinity.

### Step-by-Step Solution

1. **Identify the dominant terms**:
   - As \(x \to \infty\), the dominant term in the numerator \(\sqrt{8x^2 + 7}\) is \(\sqrt{8x^2} = 2\sqrt{2}x\).
   - The dominant term in the denominator \(3 - 2x\) is \(-2x\).

2. **Simplify the expression**:
   Divide both the numerator and the denominator by \(x\) to simplify:

   \[
   \frac{\sqrt{8x^2 + 7}}{3 - 2x} = \frac{\frac{\sqrt{8x^2 + 7}}{x}}{\frac{3}{x} - 2}.
   \]

3. **Simplify further**:
   - For the numerator: \(\frac{\sqrt{8x^2 + 7}}{x} = \sqrt{8 + \frac{7}{x^2}}\).
   - For the denominator: \(\frac{3}{x} - 2 \to -2\) as \(x \to \infty\).

   So the expression simplifies to:

   \[
   \frac{\sqrt{8 + \frac{7}{x^2}}}{-2}.
   \]

4. **Take the limit** as \(x \to \infty\):
   As \(x \to \infty\), \(\frac{7}{x^2} \to 0\), so the expression inside the square root becomes:

   \[
   \sqrt{8 + 0} = \sqrt{8} = 2\sqrt{2}.
   \]

   Therefore, the limit is:

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

### Final Answer

\[
\lim_{x \to \infty} \frac{\sqrt{8x^2 + 7}}{3 - 2x} = -\sqrt{2}.
\]

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

### Related Questions

1. How would the limit change if the denominator was \(3 + 2x\)?
2. What happens if the numerator was \(\sqrt{8x^2 - 7}\) instead of \(\sqrt{8x^2 + 7}\)?
3. Can you find the limit as \(x \to -\infty\) for the same expression?
4. How does the presence of the square root affect the rate at which the numerator grows compared to the denominator?
5. How would you evaluate \(\lim_{x \to \infty} \frac{x^2 + 7}{3 - 2x}\)?

### Tip

When taking limits as \(x \to \infty\), dividing all terms by the highest power of \(x\) in the denominator often helps to simplify the expression and identify the behavior of the limit.

Learn how to evaluate the limit of sqrt(8x^2 + 7) / (3 - 2x) as x approaches infinity with a step-by-step solution. Understand the behavior of square roots and fractions involving infinity. Suitable for advanced high school students and beyond.

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