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The given problem involves calculating the limit:

\[
\lim_{{x \to 2}} \frac{4 - x^2}{3 - \sqrt{x^2} + 5}
\]

Let's solve this step by step.

### Step 1: Simplify the expression

First, observe the function:
\[
f(x) = \frac{4 - x^2}{3 - \sqrt{x^2} + 5}
\]

The square root term simplifies as \( \sqrt{x^2} = |x| \), which for \( x = 2 \), becomes \( |2| = 2 \). So the denominator becomes:

\[
3 - \sqrt{x^2} + 5 = 3 - 2 + 5 = 6
\]

Thus, the expression simplifies to:

\[
\frac{4 - x^2}{6}
\]

### Step 2: Evaluate the limit as \( x \to 2 \)

Now, substitute \( x = 2 \) into the simplified expression:

\[
\frac{4 - 2^2}{6} = \frac{4 - 4}{6} = \frac{0}{6} = 0
\]

### Final Answer:
\[
\lim_{{x \to 2}} \frac{4 - x^2}{3 - \sqrt{x^2} + 5} = 0
\]

Would you like further clarification on any of the steps or more details? Here are five related questions that might help deepen your understanding:

1. How does the absolute value affect the simplification of \(\sqrt{x^2}\)?
2. What happens if you evaluate the limit as \(x \to -2\)?
3. Can L'Hopital's Rule be applied to limits of this type?
4. How do discontinuities affect limit calculations?
5. What are some techniques to simplify complex limits involving square roots?

**Tip:** Always check for simplifications in both the numerator and the denominator before applying limit laws.

lim (x → 2) (4 - x^2) / (3 - √x^2 + 5)

Solve the limit lim (x → 2) (4 - x^2) / (3 - √x^2 + 5) by simplifying square roots and applying substitution. A step-by-step explanation is provided, suitable for high school students in Grades 11-12.

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