https://file.aiquickdraw.com/mathbot/file/1727224494569mmhevbbu.jpg

The limit in the image can be written as:

\[
\lim_{x \to \infty} \left( \sqrt{9x^2 - 6x + 2} - (3x - 5) \right)
\]

We can simplify this step by step.

### Step 1: Simplify the square root term

We begin by focusing on the square root expression \(\sqrt{9x^2 - 6x + 2}\). As \(x\) approaches infinity, the term \(9x^2\) dominates because it grows much faster than the linear term \(-6x\) or the constant \(2\).

Thus, for large \(x\), we approximate the expression inside the square root as:
\[
\sqrt{9x^2 - 6x + 2} \approx \sqrt{9x^2} = 3x
\]

However, this approximation isn't exact, so we'll handle it rigorously by factoring \(x^2\) from the terms inside the square root:
\[
\sqrt{9x^2 - 6x + 2} = x\sqrt{9 - \frac{6}{x} + \frac{2}{x^2}}
\]

As \(x \to \infty\), the fractions involving \(x\) become very small, so:
\[
\sqrt{9 - \frac{6}{x} + \frac{2}{x^2}} \to \sqrt{9} = 3
\]

Thus, for large \(x\):
\[
\sqrt{9x^2 - 6x + 2} \approx x \cdot 3 = 3x
\]

### Step 2: Subtract the second term

Now subtract the second term from the expression:
\[
3x - (3x - 5) = 3x - 3x + 5 = 5
\]

### Step 3: Take the limit

Since the difference simplifies to \(5\) as \(x\) approaches infinity, we conclude:
\[
\lim_{x \to \infty} \left( \sqrt{9x^2 - 6x + 2} - (3x - 5) \right) = 5
\]

Thus, the value of the limit is **5**.

Let me know if you want more details, or have any questions!

Here are some related questions to practice:

1. What is the limit of \(\lim_{x \to \infty} (\sqrt{x^2 + x + 1} - x)\)?
2. How does the square root function behave as \(x \to \infty\)?
3. What would change in the solution if the expression under the square root had a higher-degree term?
4. How can L'Hôpital's Rule be used to find limits involving indeterminate forms?
5. What is the limit \(\lim_{x \to \infty} (\sqrt{4x^2 - x} - 2x)\)?

**Tip**: Always remember to simplify complex expressions by focusing on dominant terms for large values of \(x\).

Nilai limit lim (x -> ∞) dari (√(9x^2 - 6x + 2) - (3x - 5)) adalah ....

Find the limit of the expression √(9x^2 - 6x + 2) - (3x - 5) as x approaches infinity. Learn how to simplify the square root and apply limit rules to get the final result of 5. Suitable for advanced high school math students.

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