To find the limit as \( x \) approaches \( +\infty \) for the given expression:

\[
\lim_{x \to +\infty} \left(\frac{x^3 - 3x^2 + 1}{-2x^2 + 3}\right)
\]

we can proceed by analyzing the highest power of \( x \) in the numerator and the denominator.

1. The numerator is \( x^3 - 3x^2 + 1 \), which is dominated by \( x^3 \) as \( x \) grows large.
2. The denominator is \( -2x^2 + 3 \), which is dominated by \( -2x^2 \) as \( x \) grows large.

To simplify, divide both the numerator and the denominator by \( x^2 \):

\[
\lim_{x \to +\infty} \left(\frac{x^3 - 3x^2 + 1}{-2x^2 + 3}\right) = \lim_{x \to +\infty} \left(\frac{x^3/x^2 - 3x^2/x^2 + 1/x^2}{-2x^2/x^2 + 3/x^2}\right)
\]

\[
= \lim_{x \to +\infty} \left(\frac{x - 3 + \frac{1}{x^2}}{-2 + \frac{3}{x^2}}\right)
\]

As \( x \to +\infty \):

- The term \( \frac{1}{x^2} \to 0 \).
- Thus, the expression simplifies to:

\[
\lim_{x \to +\infty} \left(\frac{x - 3}{-2}\right)
\]

This can be further simplified:

\[
= \lim_{x \to +\infty} \left(\frac{x}{-2} - \frac{3}{-2}\right)
\]

\[
= \lim_{x \to +\infty} \left(-\frac{x}{2} + \frac{3}{2}\right)
\]

As \( x \to +\infty \), \( -\frac{x}{2} \to -\infty \), so:

\[
\lim_{x \to +\infty} \left(-\frac{x}{2} + \frac{3}{2}\right) = -\infty
\]

Therefore, the limit is:

\[
\boxed{-\infty}
\]

Would you like more details or have any questions?

Here are some related questions you might find interesting:

1. What would the limit be if the highest power in the numerator and denominator were the same?
2. How does the sign of the leading coefficient in the denominator affect the limit?
3. What happens if both the numerator and the denominator are polynomials of the same degree?
4. Can you have a finite limit if the highest power in the numerator is greater than the highest power in the denominator?
5. How do you handle limits at infinity when both numerator and denominator approach zero?

**Tip:** Always look at the highest degree terms in the numerator and the denominator when evaluating limits at infinity for rational functions.

Explore the limit as x approaches infinity of the rational function (x^3 - 3x^2 + 1) / (-2x^2 + 3). Understand the detailed steps and reasoning behind evaluating limits at infinity for rational functions. Suitable for advanced high school students and beyond.

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