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The problem asks to solve the following limit:

\[
\lim_{{x \to -2}} \frac{\sqrt{1 - 4x} - \sqrt{x + 11}}{2x + 4}
\]

We are asked to find the result in fractional form. Let's solve it step by step.

### Step 1: Substituting \(x = -2\)

Substitute \(x = -2\) directly into the expression:

\[
\frac{\sqrt{1 - 4(-2)} - \sqrt{-2 + 11}}{2(-2) + 4} = \frac{\sqrt{1 + 8} - \sqrt{9}}{0} = \frac{\sqrt{9} - 3}{0}
\]

This results in the form \(\frac{0}{0}\), which is an indeterminate form. Therefore, we need to use algebraic techniques, such as rationalizing the numerator.

### Step 2: Rationalizing the numerator

We multiply both the numerator and denominator by the conjugate of the numerator:

\[
\frac{\sqrt{1 - 4x} - \sqrt{x + 11}}{2x + 4} \cdot \frac{\sqrt{1 - 4x} + \sqrt{x + 11}}{\sqrt{1 - 4x} + \sqrt{x + 11}}
\]

This simplifies the numerator using the difference of squares:

\[
= \frac{(1 - 4x) - (x + 11)}{(2x + 4)(\sqrt{1 - 4x} + \sqrt{x + 11})}
\]

Simplify the expression inside the numerator:

\[
(1 - 4x) - (x + 11) = 1 - 4x - x - 11 = -5x - 10
\]

Now, the expression becomes:

\[
\frac{-5(x + 2)}{(2x + 4)(\sqrt{1 - 4x} + \sqrt{x + 11})}
\]

### Step 3: Cancel the common factor

Notice that \(2x + 4 = 2(x + 2)\), so we can cancel out the \(x + 2\) term:

\[
= \frac{-5}{2(\sqrt{1 - 4x} + \sqrt{x + 11})}
\]

### Step 4: Substitute \(x = -2\) again

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

\[
\frac{-5}{2(\sqrt{1 - 4(-2)} + \sqrt{-2 + 11})} = \frac{-5}{2(\sqrt{9} + \sqrt{9})} = \frac{-5}{2(3 + 3)} = \frac{-5}{12}
\]

### Final Answer:

The value of the limit is:

\[
\boxed{-\frac{5}{12}}
\]

Would you like further clarification or details on any step?

Here are five related questions for practice:
1. How do you handle limits that lead to indeterminate forms?
2. What is the significance of rationalizing the numerator in limit problems?
3. Can you use L'Hôpital's Rule to solve this limit problem?
4. How do you simplify expressions involving square roots?
5. What are common techniques for solving limits involving square roots?

**Tip:** When you encounter indeterminate forms like \(\frac{0}{0}\), always look for ways to simplify or factor the expression, such as rationalizing or using L'Hôpital's Rule.

Evaluate the limit as x approaches -2 for the expression (sqrt(1-4x) - sqrt(x+11)) / (2x+4).

Explore a detailed, step-by-step solution to evaluate the limit of a complex algebraic expression involving square roots as x approaches -2, using rationalization and algebraic simplification. This guide covers essential limit evaluation techniques suitable for college students.

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