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Solve this math problem, step by step and explain each one. 

The limit to solve is:

\[
\lim_{x \to 0} \frac{x}{4 - \sqrt{x + 16}}
\]

### Step 1: Substitution of \( x = 0 \)

First, let's substitute \( x = 0 \) directly into the expression:

\[
\frac{0}{4 - \sqrt{0 + 16}} = \frac{0}{4 - \sqrt{16}} = \frac{0}{4 - 4} = \frac{0}{0}
\]

This results in an indeterminate form \(\frac{0}{0}\), which means we need to manipulate the expression further to evaluate the limit.

### Step 2: Rationalizing the Denominator

To resolve the indeterminate form, we can multiply the numerator and denominator by the conjugate of the denominator:

\[
\frac{x}{4 - \sqrt{x + 16}} \times \frac{4 + \sqrt{x + 16}}{4 + \sqrt{x + 16}} = \frac{x(4 + \sqrt{x + 16})}{(4 - \sqrt{x + 16})(4 + \sqrt{x + 16})}
\]

### Step 3: Simplifying the Expression

Using the difference of squares formula:

\[
(4 - \sqrt{x + 16})(4 + \sqrt{x + 16}) = 4^2 - (\sqrt{x + 16})^2 = 16 - (x + 16) = 16 - x - 16 = -x
\]

Thus, the expression simplifies to:

\[
\frac{x(4 + \sqrt{x + 16})}{-x}
\]

### Step 4: Cancel the \( x \)

Since \( x \) is non-zero as it approaches 0, we can cancel the \( x \) in the numerator and denominator:

\[
\frac{4 + \sqrt{x + 16}}{-1} = -(4 + \sqrt{x + 16})
\]

### Step 5: Evaluate the Limit as \( x \to 0 \)

Now, substitute \( x = 0 \) into the simplified expression:

\[
-\left(4 + \sqrt{0 + 16}\right) = -(4 + 4) = -8
\]

### Final Answer

The limit is:

\[
\lim_{x \to 0} \frac{x}{4 - \sqrt{x + 16}} = -8
\]

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

Here are some related questions you might find interesting:

1. How do you apply the difference of squares formula in limits?
2. What are indeterminate forms, and how do you resolve them?
3. Can this problem be solved using L'Hôpital's Rule? How?
4. How would the limit change if the expression under the square root was different?
5. What other techniques can be used to evaluate limits of similar functions?

**Tip:** Always consider rationalizing the denominator when dealing with limits that involve square roots to eliminate indeterminate forms.

Explore how to calculate the limit of the function (x / (4 - sqrt(x + 16))) as x approaches 0 with detailed step-by-step instructions. Learn about limits, rationalizing the denominator, and resolving indeterminate forms in calculus.

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