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We are asked to find the end behavior of the function 

\[
f(x) = (3x+1)(x-2)(3-x)^2
\]

and identify the correct choice for the limits as \(x \to \infty\) and \(x \to -\infty\).

### Step 1: Simplify the expression for large values of \(x\)
We analyze the dominant terms as \(x\) becomes large (both positively and negatively).

- **Term 1:** \( (3x+1) \sim 3x \) as \(x \to \infty\) or \(x \to -\infty\).
- **Term 2:** \( (x-2) \sim x \) as \(x \to \infty\) or \(x \to -\infty\).
- **Term 3:** \( (3 - x)^2 = (-x + 3)^2 \sim x^2 \) as \(x \to \infty\) or \(x \to -\infty\).

### Step 2: End behavior as \( x \to \infty \)
- When \(x \to \infty\), each factor simplifies:
  \[
  (3x+1) \sim 3x, \quad (x-2) \sim x, \quad (3 - x)^2 \sim x^2
  \]
  So, for large \(x\),
  \[
  f(x) \sim (3x)(x)(x^2) = 3x^4
  \]
  Since the highest power of \(x\) is even and positive, as \(x \to \infty\), \(f(x) \to \infty\).

### Step 3: End behavior as \( x \to -\infty \)
- When \(x \to -\infty\), each factor behaves as follows:
  \[
  (3x+1) \sim 3x, \quad (x-2) \sim x, \quad (3 - x)^2 = (-x+3)^2 \sim x^2
  \]
  So, for large negative \(x\),
  \[
  f(x) \sim (3x)(x)(x^2) = 3x^4
  \]
  Again, since the highest power of \(x\) is even, as \(x \to -\infty\), \(f(x) \to \infty\).

### Conclusion:
The function \(f(x)\) tends to infinity as \(x \to \infty\) and as \(x \to -\infty\). Therefore, the correct answer is:

\[
\boxed{A}
\]

Would you like further details or have any questions? Here are 5 related questions to consider:

1. How do we determine the dominant terms in a polynomial for end behavior analysis?
2. What is the significance of the degree of a polynomial in determining end behavior?
3. How would the end behavior change if a negative sign were applied to the entire function?
4. What if the degree of \(x\) were odd—how would that affect the limits?
5. Can you explain the behavior of a rational function’s end behavior using limits?

**Tip:** When analyzing limits for polynomials, focus on the highest power of \(x\) for large \(x\), as this term dictates the end behavior.

Which of the following correctly give the end behavior for f(x) = (3x+1)(x−2)(3−x)^2?

Analyze the end behavior of the polynomial function f(x) = (3x+1)(x−2)(3−x)^2 as x approaches positive and negative infinity. This problem covers key concepts in limits, polynomials, and end behavior analysis. Suitable for Grades 11-12.

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