1. **State the problem:** We need to find the value of $$Q_\infty = \lim_{t \to \infty} \frac{t^2 + 2t + 1}{4t^2}$$. 2. **Recall the limit rule for rational functions:** When $$t$$ approaches infinity, the behavior of a rational function $$\frac{P(t)}{Q(t)}$$ depends on the degrees of the polynomials $$P(t)$$ and $$Q(t)$$. 3. **Identify degrees:** The numerator $$t^2 + 2t + 1$$ is degree 2, and the denominator $$4t^2$$ is also degree 2. 4. **Divide numerator and denominator by $$t^2$$ (the highest power):** $$\lim_{t \to \infty} \frac{t^2 + 2t + 1}{4t^2} = \lim_{t \to \infty} \frac{\frac{t^2}{t^2} + \frac{2t}{t^2} + \frac{1}{t^2}}{\frac{4t^2}{t^2}} = \lim_{t \to \infty} \frac{1 + \frac{2}{t} + \frac{1}{t^2}}{4}$$ 5. **Simplify the fraction:** $$\lim_{t \to \infty} \frac{1 + \frac{2}{t} + \frac{1}{t^2}}{4} = \frac{1 + \lim_{t \to \infty} \frac{2}{t} + \lim_{t \to \infty} \frac{1}{t^2}}{4}$$ 6. **Evaluate limits of terms with $$t$$ in denominator:** Since $$\lim_{t \to \infty} \frac{2}{t} = 0$$ and $$\lim_{t \to \infty} \frac{1}{t^2} = 0$$, the expression becomes: $$\frac{1 + 0 + 0}{4} = \frac{1}{4}$$ 7. **Final answer:** $$Q_\infty = \frac{1}{4}$$ This means as $$t$$ grows very large, the value of the expression approaches $$\frac{1}{4}$$.