https://file.aiquickdraw.com/mathbot/file/1726941644403al23dkh6.jpg

The limit in the image is:

\[
\lim_{x \to \infty} \frac{\sqrt{9x^2 - 1}}{2x + 1}
\]

### Step-by-step solution:

1. **Factor the expression inside the square root:**

   In the numerator, we have \(\sqrt{9x^2 - 1}\). As \(x\) approaches infinity, we focus on the leading terms in both the numerator and denominator.

   We can factor out \(x^2\) from inside the square root:
   \[
   \sqrt{9x^2 - 1} = \sqrt{x^2(9 - \frac{1}{x^2})} = x\sqrt{9 - \frac{1}{x^2}}
   \]
   Thus, the numerator becomes:
   \[
   x\sqrt{9 - \frac{1}{x^2}}
   \]

2. **Simplify the denominator:**
   The denominator is \(2x + 1\), which can be factored as:
   \[
   2x + 1 = x(2 + \frac{1}{x})
   \]

3. **Rewrite the limit:**

   Now the limit becomes:
   \[
   \lim_{x \to \infty} \frac{x\sqrt{9 - \frac{1}{x^2}}}{x(2 + \frac{1}{x})}
   \]
   Cancel the \(x\) terms:
   \[
   \lim_{x \to \infty} \frac{\sqrt{9 - \frac{1}{x^2}}}{2 + \frac{1}{x}}
   \]

4. **Evaluate the limit as \(x \to \infty\):**

   As \(x \to \infty\), \(\frac{1}{x^2} \to 0\) and \(\frac{1}{x} \to 0\). Therefore, the expression simplifies to:
   \[
   \frac{\sqrt{9 - 0}}{2 + 0} = \frac{\sqrt{9}}{2} = \frac{3}{2}
   \]

### Final answer:
\[
\lim_{x \to \infty} \frac{\sqrt{9x^2 - 1}}{2x + 1} = \frac{3}{2}
\]

Would you like more details, or do you have any further questions?

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Here are 5 related questions to explore:
1. How does factoring help in simplifying limits involving square roots?
2. What happens when the highest powers of \(x\) in the numerator and denominator are different?
3. How do you evaluate limits involving infinity when square roots are present?
4. Can you apply L'Hôpital's Rule to this problem instead of algebraic simplification?
5. What role does asymptotic behavior play in determining limits as \(x \to \infty\)?

**Tip:** When dealing with square roots in limits, factoring out the highest power of \(x\) simplifies the problem significantly.

Find the limit: lim(x→∞) (sqrt(9x^2 - 1))/(2x + 1)

Learn how to find the limit as x approaches infinity for the expression (sqrt(9x^2 - 1))/(2x + 1). This problem covers concepts of limits, factoring, and square root properties. Suitable for pre-calculus and calculus students.

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