We are tasked with evaluating the limit:

\[
\lim_{x \to 81} \frac{\sqrt{x} - 9}{\sqrt[4]{x} - 3}
\]

### Step 1: Direct substitution
If we directly substitute \( x = 81 \) into the expression, we get:

\[
\frac{\sqrt{81} - 9}{\sqrt[4]{81} - 3} = \frac{9 - 9}{3 - 3} = \frac{0}{0}
\]

This results in an indeterminate form \( \frac{0}{0} \), so we need to simplify the expression.

### Step 2: Use algebraic manipulation (factoring and rationalization)

We'll rationalize both the numerator and the denominator.

#### Numerator: \( \sqrt{x} - 9 \)
This can be rationalized using the identity \( a^2 - b^2 = (a - b)(a + b) \). In this case:

\[
\sqrt{x} - 9 = \frac{(\sqrt{x} - 9)(\sqrt{x} + 9)}{\sqrt{x} + 9} = \frac{x - 81}{\sqrt{x} + 9}
\]

#### Denominator: \( \sqrt[4]{x} - 3 \)
We can similarly rationalize the denominator by multiplying by the conjugate. Here, we use:

\[
\sqrt[4]{x} - 3 = \frac{(\sqrt[4]{x} - 3)(\sqrt[4]{x}^3 + 3\sqrt[4]{x}^2 + 9\sqrt[4]{x} + 9)}{\sqrt[4]{x}^3 + 3\sqrt[4]{x}^2 + 9\sqrt[4]{x} + 9} = \frac{x - 81}{(\sqrt[4]{x})^3 + 3\sqrt[4]{x}^2 + 9\sqrt[4]{x} + 9}
\]

### Step 3: Substitute and cancel terms
The limit now becomes:

\[
\lim_{x \to 81} \frac{\frac{x - 81}{\sqrt{x} + 9}}{\frac{x - 81}{(\sqrt[4]{x})^3 + 3\sqrt[4]{x}^2 + 9\sqrt[4]{x} + 9}}
\]

This simplifies to:

\[
\lim_{x \to 81} \frac{(\sqrt[4]{x})^3 + 3\sqrt[4]{x}^2 + 9\sqrt[4]{x} + 9}{\sqrt{x} + 9}
\]

### Step 4: Evaluate the limit
Now, substituting \( x = 81 \) directly into the simplified expression:

- \( \sqrt{81} = 9 \)
- \( \sqrt[4]{81} = 3 \)

So the limit becomes:

\[
\frac{3^3 + 3 \cdot 3^2 + 9 \cdot 3 + 9}{9 + 9} = \frac{27 + 27 + 27 + 9}{18} = \frac{90}{18} = 5
\]

Thus, the value of the limit is:

\[
\boxed{5}
\]

Would you like more details or have any questions?

Here are 5 related questions to explore further:
1. How can rationalization help resolve indeterminate forms in limits?
2. What is the difference between using direct substitution and algebraic simplification?
3. What are other techniques to solve limits when you encounter \( \frac{0}{0} \)?
4. How do higher root functions affect limit behavior compared to square roots?
5. Can you apply L'Hopital's Rule to solve this limit? How?

**Tip:** Always check if an expression results in an indeterminate form before simplifying or applying limit theorems.

This detailed solution explains how to evaluate the limit of (√x - 9) / (4√x - 3) as x approaches 81 using algebraic manipulation, rationalization, and substitution techniques. The solution includes key mathematical concepts like indeterminate forms and algebraic simplifications. Suitable for students in Grades 11-12.

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