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The limit problem shown in the image is:

\[
\lim_{{x \to 6}} \frac{x^3 - 36x}{x^3 - 23x^2 + 102x}
\]

### Step-by-step solution:

1. **Substitute \(x = 6\)** directly into the expression:

   Numerator:
   \[
   6^3 - 36(6) = 216 - 216 = 0
   \]
   
   Denominator:
   \[
   6^3 - 23(6^2) + 102(6) = 216 - 23(36) + 102(6) = 216 - 828 + 612 = 0
   \]

   So, the expression becomes \(\frac{0}{0}\), which is an indeterminate form. This means we need to apply **L'Hôpital's Rule**.

2. **Apply L'Hôpital's Rule**:
   L'Hôpital's Rule states that if we get an indeterminate form \( \frac{0}{0} \), we can differentiate the numerator and the denominator and then take the limit again.
   
   Differentiate the numerator:
   \[
   \frac{d}{dx} (x^3 - 36x) = 3x^2 - 36
   \]
   
   Differentiate the denominator:
   \[
   \frac{d}{dx} (x^3 - 23x^2 + 102x) = 3x^2 - 46x + 102
   \]
   
3. **Substitute \(x = 6\)** into the new expression:

   Numerator:
   \[
   3(6^2) - 36 = 3(36) - 36 = 108 - 36 = 72
   \]
   
   Denominator:
   \[
   3(6^2) - 46(6) + 102 = 3(36) - 46(6) + 102 = 108 - 276 + 102 = -66
   \]
   
   So, the limit becomes:
   \[
   \lim_{{x \to 6}} \frac{72}{-66} = \frac{72}{-66} = -\frac{12}{11}
   \]

### Final Answer:
\[
\lim_{{x \to 6}} \frac{x^3 - 36x}{x^3 - 23x^2 + 102x} = -\frac{12}{11}
\]

Would you like further details or have any questions?

Here are five related questions for further practice:

1. What are the conditions under which L'Hôpital's Rule can be applied?
2. How do you handle limits that result in forms other than \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\)?
3. Can L'Hôpital's Rule be applied more than once in solving a limit problem?
4. How do you differentiate polynomials in more complex rational functions?
5. What happens if L'Hôpital’s Rule leads to a constant value after the first application?

**Tip:** Always check if substitution directly into the limit yields an indeterminate form before applying L'Hôpital's Rule.

What is the limit as x approaches 6 of (x^3 - 36x) / (x^3 - 23x^2 + 102x)?

Learn how to solve a limit problem involving (x^3 - 36x) / (x^3 - 23x^2 + 102x) as x approaches 6 using L'Hôpital's Rule. This step-by-step guide covers key calculus concepts such as limits, derivatives, and indeterminate forms.

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