We are asked to evaluate the limit:

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

### Step-by-step solution:

1. **Substitute** \( x = 81 \) into the expression:

   \[
   \frac{\sqrt{81} - 9}{4\sqrt{81} - 3} = \frac{9 - 9}{4(9) - 3} = \frac{0}{36 - 3} = \frac{0}{33} = 0
   \]

Thus, the limit is:

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

### Final Answer:
\[
\boxed{0}
\]

Do you want more details or have any questions?

Here are some related questions:

1. How does substitution work in limits?
2. What is the indeterminate form and how do we handle it in limits?
3. What are some techniques for evaluating limits that give indeterminate forms?
4. What happens when the numerator or denominator tends to zero in a limit?
5. How can limits be used to understand continuity?

**Tip**: Always check if substitution simplifies the limit before applying more complex methods like factoring or L'Hopital's rule.

This step-by-step guide explains how to evaluate the limit Lim x→81 (√x - 9) / (4√x - 3) using substitution. Learn about key concepts such as limits, square roots, and algebraic simplifications. Suitable for students in Grades 10-12.

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