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The problem asks to compute the limit:

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

### Step 1: Direct Substitution
Let's try to substitute \( x = 2 \) directly into the expression:

- Numerator: \( 4 - (2)^2 = 4 - 4 = 0 \)
- Denominator: \( 3 - \sqrt{(2)^2 + 5} = 3 - \sqrt{4 + 5} = 3 - \sqrt{9} = 3 - 3 = 0 \)

Substituting gives the indeterminate form \( \frac{0}{0} \). Therefore, we must simplify the expression further by algebraic manipulation or L'Hopital's Rule.

### Step 2: Simplify the Expression
The numerator can be factored:
\[
4 - x^2 = (2 - x)(2 + x)
\]
The expression becomes:
\[
\frac{(2 - x)(2 + x)}{3 - \sqrt{x^2 + 5}}
\]

### Step 3: Use L'Hopital's Rule
Since direct substitution leads to an indeterminate form, we can apply L'Hopital's Rule. This requires us to take the derivative of the numerator and denominator.

#### Derivative of the Numerator:
\[
\frac{d}{dx} (4 - x^2) = -2x
\]

#### Derivative of the Denominator:
\[
\frac{d}{dx} \left( 3 - \sqrt{x^2 + 5} \right) = - \frac{x}{\sqrt{x^2 + 5}}
\]

### Step 4: Apply L'Hopital's Rule
Now applying L'Hopital's Rule, we compute the new limit:

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

Simplifying the expression:

\[
\lim_{x \to 2} \frac{-2x \cdot \sqrt{x^2 + 5}}{-x} = \lim_{x \to 2} 2 \sqrt{x^2 + 5}
\]

Now substitute \( x = 2 \):

\[
2 \sqrt{2^2 + 5} = 2 \sqrt{4 + 5} = 2 \times 3 = 6
\]

### Final Answer:
The value of the limit is \( \boxed{6} \).

Let me know if you need more details, or have any questions. Here are some related questions to explore:

1. How does L'Hopital's Rule help in solving indeterminate forms?
2. Can you always apply L'Hopital's Rule, or are there restrictions?
3. What are some other methods for resolving \( \frac{0}{0} \) indeterminate forms?
4. How can you factor \( a^2 - b^2 \) to simplify expressions?
5. When does substituting values directly in limits give valid results?

**Tip**: Whenever you encounter the \( \frac{0}{0} \) form in limits, check if the expression can be factored or if L'Hopital's Rule applies.

Compute the limit: lim (x → 2) (4 - x^2) / (3 - √(x^2 + 5)).

This problem involves calculating the limit of a rational function as x approaches 2. The solution covers L'Hopital's Rule and factorization to resolve the indeterminate form 0/0, suitable for advanced high school students in Grades 11-12.

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