limit as x approaches 9 of 15 * f(x) over g(x)

We are tasked with finding the limit as \( x \) approaches 9 of the following expression:

\[
\lim_{x \to 9} \frac{15 \cdot f(x)}{g(x)}
\]

To properly evaluate this, we would typically need more information about the functions \( f(x) \) and \( g(x) \). However, here are the general steps to solve the limit, assuming no specific information is given:

### Steps:
1. **Direct Substitution**: The first approach is to substitute \( x = 9 \) directly into both \( f(x) \) and \( g(x) \). This gives \( f(9) \) and \( g(9) \), provided the functions are continuous at \( x = 9 \).

2. **Indeterminate Forms**: If direct substitution yields an indeterminate form like \( \frac{0}{0} \), then other methods like factoring, L'Hôpital's Rule, or simplifying the expressions may be necessary.

3. **L'Hôpital's Rule**: If the limit results in an indeterminate form such as \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), you can apply L'Hôpital's Rule by differentiating the numerator and denominator separately and then taking the limit.

4. **Alternative Simplification**: If there are specific values or behaviors of \( f(x) \) and \( g(x) \) near 9, use those to simplify the expression further.

Without specific values for \( f(x) \) and \( g(x) \), I can't proceed further. Do you have more details on these functions?

Let me know if you want additional explanation or specific help on handling certain limit cases!

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### 5 related questions:
1. What happens if \( f(x) = x^2 \) and \( g(x) = x-9 \)?
2. How would you apply L'Hôpital's Rule if \( f(9) = 0 \) and \( g(9) = 0 \)?
3. Can this limit be solved if \( g(x) = (x-9)^2 \)?
4. What techniques are useful when dealing with indeterminate forms in limits?
5. What are some common limits involving rational functions like \( \frac{f(x)}{g(x)} \)?

### Tip:
Always check if direct substitution gives a finite value before applying more complex methods like L'Hôpital’s Rule.

Learn how to solve the limit of 15 * f(x) over g(x) as x approaches 9. This calculus problem covers key concepts like limits, indeterminate forms, and applying L'Hôpital's Rule. Suitable for advanced high school and college-level students.

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