The limit in question is:

\[
\lim_{{x \to 2}} \frac{\sqrt{x} - 2}{\sqrt{x^2 - 4}}
\]

To evaluate this limit, let's first substitute \(x = 2\) directly into the expression:

\[
\frac{\sqrt{2} - 2}{\sqrt{2^2 - 4}} = \frac{\sqrt{2} - 2}{\sqrt{4 - 4}} = \frac{\sqrt{2} - 2}{\sqrt{0}} = \frac{\sqrt{2} - 2}{0}
\]

This results in an indeterminate form of \(\frac{0}{0}\), so we'll need to apply algebraic manipulation to resolve the indeterminate form.

### Step 1: Rationalizing the numerator
We can rationalize the numerator by multiplying the numerator and the denominator by the conjugate of the numerator, which is \(\sqrt{x} + 2\):

\[
\lim_{{x \to 2}} \frac{\sqrt{x} - 2}{\sqrt{x^2 - 4}} \cdot \frac{\sqrt{x} + 2}{\sqrt{x} + 2} = \lim_{{x \to 2}} \frac{(\sqrt{x} - 2)(\sqrt{x} + 2)}{(\sqrt{x^2 - 4})(\sqrt{x} + 2)}
\]

### Step 2: Simplify the numerator
Using the difference of squares formula \(a^2 - b^2 = (a-b)(a+b)\):

\[
\lim_{{x \to 2}} \frac{x - 4}{(\sqrt{x^2 - 4})(\sqrt{x} + 2)}
\]

### Step 3: Simplify further
Notice that \(x^2 - 4\) can be factored as \((x - 2)(x + 2)\):

\[
\lim_{{x \to 2}} \frac{x - 4}{\sqrt{(x-2)(x+2)} \cdot (\sqrt{x} + 2)} = \lim_{{x \to 2}} \frac{(x-2)(x+2)}{\sqrt{(x-2)(x+2)} \cdot (\sqrt{x} + 2)}
\]

We can cancel out the \((x-2)\) term:

\[
\lim_{{x \to 2}} \frac{\sqrt{x+2}}{\sqrt{x} + 2}
\]

### Step 4: Substitute \(x = 2\) now:

\[
\frac{\sqrt{2+2}}{\sqrt{2} + 2} = \frac{\sqrt{4}}{\sqrt{2} + 2} = \frac{2}{\sqrt{2} + 2}
\]

### Step 5: Rationalize the denominator if necessary
Multiply the numerator and denominator by the conjugate of the denominator \(\sqrt{2} - 2\):

\[
\frac{2(\sqrt{2} - 2)}{(\sqrt{2} + 2)(\sqrt{2} - 2)} = \frac{2(\sqrt{2} - 2)}{2 - 4} = \frac{2(\sqrt{2} - 2)}{-2} = \sqrt{2} - 2
\]

Thus, the limit is:

\[
\lim_{{x \to 2}} \frac{\sqrt{x} - 2}{\sqrt{x^2 - 4}} = \boxed{\sqrt{2} - 2}
\]

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

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Here are 5 related questions:
1. How do you simplify limits involving square roots?
2. What is the difference of squares, and how is it used in limits?
3. How do you rationalize a denominator in a fraction?
4. What are common techniques for resolving indeterminate forms in limits?
5. How does substituting values directly help in evaluating limits?

**Tip:** When dealing with limits that result in indeterminate forms, always consider algebraic manipulation, such as factoring, rationalizing, or using L'Hôpital's rule.

Learn how to evaluate the limit lim x→2 √x-2/√x²-4 with a step-by-step solution. This problem involves key concepts like limits, square roots, and rationalization, and is suitable for students in Grades 11-12.

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