Evaluate the limit
lim𝑎→16 16−𝑎/4−sqaure root 𝑎= 


We are tasked with evaluating the limit:

\[
\lim_{a \to 16} \frac{16 - a}{4 - \sqrt{a}}
\]

Substituting \( a = 16 \) directly gives an indeterminate form \(\frac{0}{0}\). Therefore, we need to manipulate the expression to resolve this indeterminate form, typically by rationalizing the denominator.

### Step 1: Multiply by the conjugate of the denominator
To simplify, multiply the numerator and denominator by the conjugate of the denominator \(4 + \sqrt{a}\):

\[
\frac{16 - a}{4 - \sqrt{a}} \cdot \frac{4 + \sqrt{a}}{4 + \sqrt{a}} = \frac{(16 - a)(4 + \sqrt{a})}{(4 - \sqrt{a})(4 + \sqrt{a})}
\]

The denominator simplifies using the difference of squares:

\[
(4 - \sqrt{a})(4 + \sqrt{a}) = 16 - a
\]

So the expression becomes:

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

### Step 2: Cancel the common factor
Since \(16 - a \neq 0\) for \(a \neq 16\), we can cancel \(16 - a\) from the numerator and denominator:

\[
4 + \sqrt{a}
\]

### Step 3: Substitute \(a = 16\)
Now, substitute \(a = 16\) into the simplified expression:

\[
4 + \sqrt{16} = 4 + 4 = 8
\]

Thus, the limit is:

\[
\boxed{8}
\]

Would you like further clarification or details?

Here are 5 related questions you might want to explore:
1. What are the different types of indeterminate forms in limits?
2. How do you rationalize a denominator with square roots?
3. What is L'Hopital's rule, and when should it be applied?
4. How can you solve limits involving higher powers of \(a\)?
5. How does the difference of squares simplify algebraic expressions?

**Tip:** When encountering indeterminate forms like \(0/0\), always look for algebraic manipulations like factoring or rationalization before applying L'Hopital’s rule.

Evaluate the limit lim𝑎→16 (16−𝑎)/(4−√𝑎)

Learn how to evaluate the limit lim𝑎→16 (16−𝑎)/(4−√𝑎) by rationalizing the denominator. This problem introduces concepts such as limits, difference of squares, and indeterminate forms, with a step-by-step solution for students in Grades 11-12 or early calculus courses.

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