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The problem asks us to evaluate the following limit:

\[
\lim_{x \to 4} \frac{x - 4}{\sqrt{x^2 - 16}}
\]

### Step 1: Simplify the expression

We recognize that \( x^2 - 16 \) can be factored as a difference of squares:

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

Thus, the expression becomes:

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

### Step 2: Handle the limit as \( x \to 4 \)

If we directly substitute \( x = 4 \) into this simplified expression, we get:

\[
\frac{4 - 4}{\sqrt{(4 - 4)(4 + 4)}} = \frac{0}{\sqrt{0 \cdot 8}} = \frac{0}{0}
\]

This is an indeterminate form, so we need to simplify the expression further.

### Step 3: Use algebraic manipulation to resolve the indeterminate form

To resolve the indeterminate form, we can try a substitution. Let’s factor the numerator and denominator, and simplify the expression:

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

### Step 4: Evaluate the limit

Now, substitute \( x = 4 \) into the simplified expression:

\[
\frac{1}{\sqrt{4 + 4}} = \frac{1}{\sqrt{8}} = \frac{1}{2\sqrt{2}}
\]

Multiplying the numerator and denominator by \( \sqrt{2} \), we get:

\[
\frac{1}{2\sqrt{2}} = \frac{\sqrt{2}}{4}
\]

However, this is not one of the provided answer choices. Upon revisiting, it seems like something was missed during simplification. Let's backtrack or recheck the given question structure. 

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Evaluate the limit: lim_{x -> 4} (x - 4) / sqrt(x^2 - 16).

Learn how to evaluate the limit lim_{x -> 4} (x - 4) / sqrt(x^2 - 16). This step-by-step guide uses algebraic simplification, focusing on factoring and resolving indeterminate forms. Ideal for high school students preparing for calculus topics.

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