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Let's analyze the given limit problem:

\[
\lim_{{x \to 4}} \frac{{3 - x}}{{(x - 4)^2}}
\]

We are asked to determine whether this limit goes to \( +\infty \) or \( -\infty \) as \( x \to 4 \).

### Step-by-step analysis:

1. **Simplifying the numerator**:
   \[
   3 - x = -(x - 4)
   \]
   This is a simple rearrangement to rewrite the numerator in terms of \( (x - 4) \).

2. **Substituting the rearranged expression**:
   \[
   \lim_{{x \to 4}} \frac{{-(x - 4)}}{{(x - 4)^2}}
   \]

3. **Simplifying the expression**:
   \[
   \lim_{{x \to 4}} \frac{{-(x - 4)}}{{(x - 4)(x - 4)}} = \lim_{{x \to 4}} \frac{{-1}}{{x - 4}}
   \]

4. **Analyzing the behavior as \( x \to 4 \)**:
   - When \( x \to 4^+ \) (from the right side), \( x - 4 \) is positive, so \( \frac{{-1}}{{x - 4}} \to -\infty \).
   - When \( x \to 4^- \) (from the left side), \( x - 4 \) is negative, so \( \frac{{-1}}{{x - 4}} \to +\infty \).

Since the question asks for the **infinite limit**, and from both sides, the expression tends toward infinity (but of different signs), the correct choice is **\( -\infty \)**.

Thus, the answer is:

\[
\boxed{-\infty}
\]

Would you like further clarification on any step? Here are some related questions:

1. How do we handle limits involving indeterminate forms?
2. What is the general approach for limits involving rational functions?
3. How can you determine if a limit approaches \( +\infty \) or \( -\infty \)?
4. What are the differences between one-sided and two-sided limits?
5. How would this limit change if the power in the denominator were different?

**Tip:** When working with limits that involve division by small numbers, always consider approaching the limit from both sides (left-hand and right-hand limits) to ensure accurate results.

Determine the infinite limit: lim (x→4) (3 - x) / (x - 4)^2

This page explains how to evaluate the infinite limit of (3 - x) / (x - 4)^2 as x approaches 4. Detailed steps include simplifying the expression and analyzing the behavior of the function from both sides. Suitable for high school and college calculus students.

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