The problem involves finding the limit:

$\lim_{x \to 3} \frac{f(x) - f(3)}{x - 3}$

Given that $f(x) = \ln(x)$, this expression can be identified as the definition of the derivative of $f(x)$ at $x = 3$.

Steps:

1. Definition of derivative:

   The derivative of a function $f(x)$ at a point $a$ is given by:

   $f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}$

   Here, $f(x) = \ln(x)$ and $a = 3$. So the given limit represents $f'(3)$.

2. Derivative of $f(x) = \ln(x)$:

   We know that the derivative of $\ln(x)$ is:

   $f'(x) = \frac{1}{x}$

3. Evaluating at $x = 3$:

   $f'(3) = \frac{1}{3}$

Thus, the limit is:

$\lim_{x \to 3} \frac{\ln(x) - \ln(3)}{x - 3} = \frac{1}{3}$

Final answer:

$\boxed{\frac{1}{3}}$

Would you like further details or have any questions about this problem?

Here are 5 related questions for further exploration:

1. How would you compute the derivative of $\ln(x)$ using first principles?
2. What is the geometric interpretation of this limit?
3. How does this problem change if $f(x)$ were a different logarithmic function?
4. Can you apply L'Hôpital's Rule to solve this limit?
5. How would you calculate the derivative of $\ln(x)$ at any arbitrary point $a$?

Tip: When faced with a limit resembling the difference quotient, always check if it represents the derivative of a function at a specific point.