To evaluate the limit

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

we can start by simplifying the expression under the square root and identifying the dominant terms as \(x\) approaches infinity.

### Step-by-Step Solution

1. **Rewrite the expression:**
   We start by factoring out \(x^2\) inside the square root:

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

2. **Simplify the expression:**
   The expression now is:

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

   Factor \(x\) out from the entire expression:

   \[
   x\left(\frac{\sqrt{8 + \frac{7}{x^2}}}{3} - 2\right).
   \]

3. **Take the limit as \(x \to \infty\):**
   As \(x \to \infty\), \(\frac{7}{x^2} \to 0\). So, the square root term \(\sqrt{8 + \frac{7}{x^2}}\) approaches \(\sqrt{8}\).

   Therefore,

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

4. **Substitute this into the limit expression:**
   The limit now simplifies to:

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

   Simplify the coefficient:

   \[
   \frac{2\sqrt{2}}{3} - 2 = \frac{2\sqrt{2}}{3} - \frac{6}{3} = \frac{2\sqrt{2} - 6}{3}.
   \]

5. **Analyze the coefficient:**
   Since \(2\sqrt{2} \approx 2.828\), we have \(2\sqrt{2} - 6 < 0\). Thus, the coefficient is negative.

6. **Determine the final limit:**
   As \(x \to \infty\), the expression \(x \left(\frac{2\sqrt{2} - 6}{3}\right)\) goes to \(-\infty\) because the coefficient is negative.

### Final Answer

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

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

### Related Questions

1. How would the limit change if the square root expression was \(\sqrt{8x^2 - 7}\)?
2. What happens to the limit if the constant \(2x\) is replaced by \(3x\)?
3. Can you calculate the limit if \(x \to -\infty\) for the same expression?
4. How does this limit relate to limits involving logarithmic functions?
5. How would you evaluate \(\lim_{x \to \infty} \frac{\sqrt{8x^2 + 7} - 2x}{x}\)?

### Tip

Always consider factoring out the dominant term in the expression when dealing with limits as \(x \to \infty\) to simplify the analysis.

Learn how to evaluate the limit lim x->∞ sqrt(8x^2 + 7) / (3 - 2x) with a detailed step-by-step solution. Understand the behavior of square roots and linear expressions as x approaches infinity. Suitable for advanced high school students and anyone studying calculus.

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