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The problem asks for the value of the limit:

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

Let's solve it step by step.

### Step 1: Factor the numerator
We need to factor the quadratic expression in the numerator, \(x^2 - 3x + 2\).

\[
x^2 - 3x + 2 = (x - 1)(x - 2).
\]

So, the original expression becomes:

\[
\lim_{x \to 2} \frac{(x - 1)(x - 2)}{\sqrt{x} - \sqrt{2}}.
\]

### Step 2: Apply the limit
When we directly substitute \(x = 2\), we get an indeterminate form \( \frac{0}{0} \), so we need to simplify further.

### Step 3: Rationalize the denominator
To eliminate the square root in the denominator, we multiply both the numerator and denominator by the conjugate of the denominator, which is \(\sqrt{x} + \sqrt{2}\):

\[
\frac{(x - 1)(x - 2)}{\sqrt{x} - \sqrt{2}} \times \frac{\sqrt{x} + \sqrt{2}}{\sqrt{x} + \sqrt{2}}.
\]

This simplifies the expression to:

\[
\lim_{x \to 2} \frac{(x - 1)(x - 2)(\sqrt{x} + \sqrt{2})}{x - 2}.
\]

### Step 4: Cancel common factors
The factor \(x - 2\) appears in both the numerator and denominator, so we can cancel it:

\[
\lim_{x \to 2} (x - 1)(\sqrt{x} + \sqrt{2}).
\]

### Step 5: Evaluate the limit
Now, substitute \(x = 2\):

\[
(2 - 1)(\sqrt{2} + \sqrt{2}) = 1 \times 2\sqrt{2} = 2\sqrt{2}.
\]

### Final Answer:
The value of the limit is \(2\sqrt{2}\).

---

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

Here are 5 questions related to this:

1. What other methods can be used to evaluate indeterminate limits like \(\frac{0}{0}\)?
2. How do you factor a quadratic expression like \(x^2 - 3x + 2\)?
3. Why do we multiply by the conjugate when simplifying square roots in limits?
4. What are the common forms of indeterminate limits, and how are they handled?
5. Can L'Hopital's Rule be applied to this limit? If so, how?

**Tip:** Always check for indeterminate forms before applying algebraic simplifications when solving limits.

Nilai dari lim x → 2 (x^2 - 3x + 2) / sqrt(x - 2) adalah?

Explore the solution for evaluating the limit lim x → 2 (x^2 - 3x + 2) / sqrt(x - 2) using factorization and rationalization techniques. This detailed guide covers the key steps in solving limits involving quadratic expressions and square roots, perfect for high school students.

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