1. **Problem:** Find the limit $$\lim_{n \to +\infty} \frac{4n^2}{1 - 3n^2}$$. 2. **Formula and rules:** When evaluating limits of rational functions as $$n$$ approaches infinity, divide numerator and denominator by the highest power of $$n$$ in the denominator to simplify. 3. **Step-by-step solution:** - The highest power of $$n$$ in the denominator is $$n^2$$. - Divide numerator and denominator by $$n^2$$: $$\lim_{n \to +\infty} \frac{4n^2 / n^2}{(1 - 3n^2) / n^2} = \lim_{n \to +\infty} \frac{4}{\frac{1}{n^2} - 3}$$ - As $$n \to +\infty$$, $$\frac{1}{n^2} \to 0$$, so the limit becomes: $$\frac{4}{0 - 3} = \frac{4}{-3} = -\frac{4}{3}$$ 4. **Interpretation:** The limit is the ratio of the leading coefficients of the highest degree terms, with the denominator's sign considered. **Final answer:** $$\boxed{-\frac{4}{3}}$$