https://file.aiquickdraw.com/mathbot/file/1727066791557c3m1es4o.jpg

The image shows a mathematical limit problem. Here's the expression I can extract from the image:

\[
\lim_{x \to 1} \frac{3x}{\sqrt{9x} - 3}
\]

To solve this limit, let's first try substituting \( x = 1 \) directly into the expression:

\[
\frac{3(1)}{\sqrt{9(1)} - 3} = \frac{3}{\sqrt{9} - 3} = \frac{3}{3 - 3} = \frac{3}{0}
\]

This results in division by zero, which means we need to apply a different technique, such as rationalizing the denominator.

### Steps:
1. **Multiply the numerator and denominator** by the conjugate of the denominator: \( \sqrt{9x} + 3 \). This helps eliminate the square root.

2. Let's express the new limit after multiplying:
   \[
   \lim_{x \to 1} \frac{3x(\sqrt{9x} + 3)}{(\sqrt{9x} - 3)(\sqrt{9x} + 3)}
   \]

3. The denominator simplifies using the difference of squares:
   \[
   (\sqrt{9x})^2 - 3^2 = 9x - 9
   \]

4. The new expression becomes:
   \[
   \lim_{x \to 1} \frac{3x(\sqrt{9x} + 3)}{9(x - 1)}
   \]

Now, I will simplify and evaluate this expression.

Let me calculate this limit.The limit evaluates to infinity (\(\infty\)). This suggests that the function grows without bound as \(x\) approaches 1 from both sides.

Would you like more details on how the limit was handled or how this type of problem can be generalized?

Here are some related questions to explore:
1. What is the difference between limits approaching finite values and infinity?
2. How can we handle indeterminate forms such as \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\) in limits?
3. How does rationalizing the denominator help solve limits involving square roots?
4. What are some other techniques for solving difficult limits, such as L'Hopital's Rule?
5. How can limits at infinity differ from limits at a specific point?

**Tip:** Whenever direct substitution in a limit leads to an indeterminate form like \(\frac{0}{0}\), rationalizing or using L'Hopital's Rule are useful strategies for further analysis.

Evaluate the limit \( \lim_{x \to 1} \frac{3x}{\sqrt{9x} - 3} \)

Learn how to evaluate the limit \( \lim_{x \to 1} \frac{3x}{\sqrt{9x} - 3} \) using rationalization. This problem explores the use of limit laws and algebraic techniques, such as the difference of squares and handling indeterminate forms.

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