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Let's analyze the mathematical expression and complete the solution step-by-step.

The given problem deals with finding the limit:
\[
m_{\text{tan}} = \lim_{x \to 16} \frac{\sqrt{x} - 4}{( \sqrt{x} - 4)( \sqrt{x} + 4 )}
\]

**Step 1: Simplify the expression**

Using the factorization \( x - 16 = (\sqrt{x} - 4)(\sqrt{x} + 4) \), the given expression becomes:
\[
m_{\text{tan}} = \lim_{x \to 16} \frac{\sqrt{x} - 4}{(\sqrt{x} - 4)(\sqrt{x} + 4)}
\]
We can cancel out the common factor \((\sqrt{x} - 4)\), assuming \(x \neq 16\):
\[
m_{\text{tan}} = \lim_{x \to 16} \frac{1}{\sqrt{x} + 4}
\]

**Step 2: Evaluate the limit**

Substituting \(x = 16\) into the simplified expression:
\[
m_{\text{tan}} = \frac{1}{\sqrt{16} + 4} = \frac{1}{4 + 4} = \frac{1}{8}
\]

Therefore, the answer in the blank is \(\frac{1}{8}\).

Would you like further details or explanations?

Here are 5 related questions to expand on this topic:

1. How do you apply limits to solve indeterminate forms?
2. What are some common factorization techniques used in calculus?
3. How is the derivative related to the concept of finding the slope of a tangent line?
4. How do you handle other forms of indeterminate expressions, such as \( \frac{0}{0} \)?
5. What are the different rules for evaluating limits when direct substitution fails?

**Tip:** Always check if factorization or simplification can resolve an indeterminate form before applying L'Hospital's Rule.

Using the factorization x − 16 = (√x − 4)(√x + 4), find the value of the limit as x approaches 16.

Learn how to solve a limit problem using algebraic factorization. This step-by-step guide demonstrates how to simplify the expression and evaluate the limit as x approaches 16, using the factorization x - 16 = (√x - 4)(√x + 4). Perfect for students in Grades 11-12.

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