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We are tasked with finding the limit:

\[
\lim_{{x \to \infty}} \frac{x^3 - 2x + 3}{4 - 4x^2}
\]

### Step-by-Step Solution:

1. **Identify the dominant terms:**
   - In the numerator, \(x^3 - 2x + 3\), the term with the highest degree is \(x^3\).
   - In the denominator, \(4 - 4x^2\), the term with the highest degree is \(-4x^2\).

2. **Divide both the numerator and denominator by \(x^3\) (the highest degree term in the numerator):**

   \[
   \frac{x^3 - 2x + 3}{4 - 4x^2} = \frac{\frac{x^3}{x^3} - \frac{2x}{x^3} + \frac{3}{x^3}}{\frac{4}{x^3} - \frac{4x^2}{x^3}}
   \]

   Simplifying this:

   \[
   = \frac{1 - \frac{2}{x^2} + \frac{3}{x^3}}{\frac{4}{x^3} - \frac{4}{x}}
   \]

3. **Evaluate the limit as \(x \to \infty\):**
   - As \(x \to \infty\), \(\frac{2}{x^2} \to 0\), \(\frac{3}{x^3} \to 0\), \(\frac{4}{x^3} \to 0\), and \(\frac{4}{x} \to 0\).
   
   This leaves us with:

   \[
   \frac{1 - 0 + 0}{0 - 0} = \frac{1}{0}
   \]

4. **Conclusion:**
   Since the limit tends toward \(\frac{1}{0}\), this implies the limit approaches infinity.

Thus, the limit is:

\[
\lim_{{x \to \infty}} \frac{x^3 - 2x + 3}{4 - 4x^2} = \infty
\]

Would you like further details or explanations on any specific part?

Here are some related questions:
1. What happens if we change the highest degree term in the denominator?
2. How would the limit change if the numerator was of a higher degree than the denominator?
3. What is the significance of dominant terms in limits involving polynomials?
4. How can we apply L'Hopital's Rule in cases like this one?
5. How does dividing by the highest power of \(x\) help in evaluating limits at infinity?

**Tip:** When calculating limits at infinity for rational functions, always compare the degrees of the polynomials in the numerator and denominator to find the dominant terms.

Find the limit as x approaches infinity: (x^3 - 2x + 3) / (4 - 4x^2).

Understand how to calculate the limit as x approaches infinity for the rational function (x^3 - 2x + 3) / (4 - 4x^2). This problem uses concepts of polynomial division and rational function behavior to determine that the limit approaches infinity. Suitable for advanced high school or college calculus students.

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