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.