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Given that:

\[
\lim_{x \to 2} \left( \frac{3f(x) - 1}{x - 3} \right) = 4,
\]

we need to determine:

\[
\lim_{x \to 2} f(x).
\]

Let's analyze this problem step-by-step.

### Step 1: Understanding the Limit Condition
The limit of a function as \( x \to 2 \) is given by:

\[
\lim_{x \to 2} \left( \frac{3f(x) - 1}{x - 3} \right) = 4.
\]

This implies that as \( x \) approaches 2, the value of \( \frac{3f(x) - 1}{x - 3} \) approaches 4. To avoid an indeterminate form, the numerator \( 3f(x) - 1 \) must balance with the denominator \( x - 3 \). As \( x \to 2 \), the denominator \( x - 3 \) approaches \( -1 \).

### Step 2: Setting Up the Equation
If \( x \) approaches 2, then:

\[
\lim_{x \to 2} \frac{3f(2) - 1}{2 - 3} = 4.
\]

Simplifying the denominator, we get:

\[
\frac{3f(2) - 1}{-1} = 4.
\]

### Step 3: Solving for \( f(2) \)
Simplifying:

\[
3f(2) - 1 = -4.
\]

Adding 1 to both sides:

\[
3f(2) = -3.
\]

Dividing by 3:

\[
f(2) = -1.
\]

### Final Answer
\[
\lim_{x \to 2} f(x) = -1.
\]

Would you like more details or have any further questions?

### Related Questions
1. How can you verify if a limit exists for a given function?
2. What happens if \( f(2) \) was undefined? How would the result change?
3. How do you apply L'Hospital's Rule to solve limits like this?
4. How would the result change if the limit was \( \lim_{x \to 3} \)?
5. Can we generalize this method for other types of rational limits?

### Tip
When dealing with limits of rational functions, always check if the numerator and denominator both approach zero or infinity to decide if L'Hospital's Rule can be applied.

Suppose lim(x → 2) ((3f(x) - 1) / (x - 3)) equals 4. What is lim(x → 2) f(x)?

Learn how to find the limit of f(x) as x approaches 2, given lim(x → 2) ((3f(x) - 1) / (x - 3)) equals 4. Step-by-step solution exploring algebraic methods and limit theorems, ideal for high school students in Grades 11-12.

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