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To solve the problem, we need to evaluate the following limit:

\[
\lim_{x \to 3} \frac{3x^2 - 9x}{f(2x) - 1}
\]

Given that \(f(x)\) is a piecewise function shown in the graph, we should first determine the values of \(f(2x)\) near \(x = 3\).

### Step 1: Simplify the numerator
The numerator \(3x^2 - 9x\) can be factored:

\[
3x^2 - 9x = 3x(x - 3)
\]

### Step 2: Evaluate the denominator \(f(2x) - 1\) at \(x = 3\)
Since we are approaching \(x = 3\), we calculate \(2x\) at \(x = 3\):

\[
2(3) = 6
\]

Now, find the value of \(f(6)\) from the graph. Looking at the graph, \(f(6) = 4\).

So, the denominator \(f(2x) - 1\) at \(x = 3\) becomes:

\[
f(6) - 1 = 4 - 1 = 3
\]

### Step 3: Substitute and evaluate the limit
Substituting the values we have:

\[
\lim_{x \to 3} \frac{3x(x - 3)}{3}
\]

This simplifies to:

\[
\lim_{x \to 3} x(x - 3)
\]

Since \(x - 3\) becomes 0 as \(x\) approaches 3, the limit evaluates to:

\[
3(3 - 3) = 3 \times 0 = 0
\]

Thus, the value of the limit is:

\[
\boxed{0}
\]

Would you like any additional details on this problem? Here are some related questions you might find useful:

1. What are the steps to evaluate limits involving piecewise functions?
2. How does factoring help in simplifying limits?
3. What is the significance of one-sided limits in evaluating functions?
4. How can we determine the continuity of a piecewise function?
5. How does the graph of a function influence the calculation of limits?

**Tip:** Always check the function values at the limit point before simplifying expressions to avoid mistakes in evaluating the limit.

Determine the value of lim(x→3) (3x^2 - 9x) / (f(2x) - 1) based on the graph of f(x).

Learn how to evaluate lim(x→3) (3x^2 - 9x) / (f(2x) - 1) using graph interpretation and algebraic factoring. This detailed step-by-step solution simplifies the numerator and uses the graph of f(x) for the denominator. Suitable for Grades 11-12.

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