1. Let's state the problem: We want to analyze the expression $$\frac{2n^2 - 3}{n^2 + 3}$$ and explain why this expression is limited (bounded). 2. First, observe the expression's numerator and denominator: - Numerator: $$2n^2 - 3$$ - Denominator: $$n^2 + 3$$ 3. Since $$n^2$$ is always non-negative for all real $$n$$, the denominator $$n^2 + 3$$ is always positive and never zero. This means the expression is defined for all real $$n$$. 4. To understand if the expression is limited (bounded), we analyze the behavior as $$|n|$$ becomes very large (goes to infinity). The highest degree terms dominate: $$\frac{2n^2 - 3}{n^2 + 3} \approx \frac{2n^2}{n^2} = 2$$ 5. Next, let's consider the precise form by dividing numerator and denominator by $$n^2$$ (assuming $$n \neq 0$$): $$\frac{2n^2 - 3}{n^2 + 3} = \frac{2 - \frac{3}{n^2}}{1 + \frac{3}{n^2}}$$ 6. As $$n \to \pm \infty$$, $$\frac{3}{n^2} \to 0$$, so the expression approaches: $$\frac{2 - 0}{1 + 0} = 2$$ 7. To check if the expression can take values beyond some bounds, simplify the expression differently. Let's denote $$y = \frac{2n^2 - 3}{n^2 + 3}$$. 8. Solve for $$n^2$$ in terms of $$y$$: $$y(n^2 + 3) = 2n^2 - 3$$ $$y n^2 + 3y = 2n^2 - 3$$ $$2n^2 - y n^2 = 3y + 3$$ $$(2 - y) n^2 = 3(y + 1)$$ $$n^2 = \frac{3(y + 1)}{2 - y}$$ 9. For $$n^2$$ to be real and non-negative, the right side must be $$\geq 0$$: $$\frac{3(y + 1)}{2 - y} \geq 0$$ 10. Analyze the inequality for $$y$$: - Numerator: $$3(y + 1)$$ has the same sign as $$y + 1$$. - Denominator: $$2 - y$$ has the same sign as $$2 - y$$. 11. The fraction is non-negative when numerator and denominator share the same sign: - Case 1: $$y + 1 \geq 0$$ and $$2 - y > 0$$ $$y \geq -1$$ and $$y < 2$$ - Case 2: $$y + 1 \leq 0$$ and $$2 - y < 0$$ $$y \leq -1$$ and $$y > 2$$ which is impossible 12. Therefore, $$y$$ must satisfy $$-1 \leq y < 2$$. 13. Conclusion: The expression $$\frac{2n^2 - 3}{n^2 + 3}$$ is limited/bounded between $$-1$$ and $$2$$, never reaching $$2$$ but approaching it asymptotically, and attaining $$-1$$ at some points. 14. This boundedness occurs because numerator and denominator are quadratic polynomials with leading terms of the same degree, and the limits of the quotient are finite as $$n$$ grows large. Final answer: The expression is limited because its values lie in the range $$-1 \leq \frac{2n^2 - 3}{n^2 + 3} < 2$$.