Server Load Response Latency (Computer Science) Solution

Part 1: Dominance Filtering

For large $N$ , the quadratic terms $N^2$ grow much faster than linear terms $N$ or constants.

Part 2: Analytical Limit Calculation

Divide every term by the highest denominator power $N^2$ :

\lim_{N \to \infty} \frac{\frac{12N^2}{N^2} + \frac{45N}{N^2}}{\frac{3N^2}{N^2} + \frac{8}{N^2}} = \lim_{N \to \infty} \frac{12 + \frac{45}{N}}{3 + \frac{8}{N^2}}

As $N \to \infty$ , the reciprocal terms approach zero:

\frac{12 + 0}{3 + 0} = \frac{12}{3} = 4\text{ ms}

Part 3: Horizontal Asymptote & Systems Architecture Conclusion

Horizontal Asymptote: $y = 4$
System SLA Impact: As the cluster scales indefinitely, the average request latency levels off at a deterministic floor of $4\text{ ms}$ . This informs cloud engineers that infrastructure latency will not degrade beyond $4\text{ ms}$ under heavy node concurrency, satisfying strict real-time user experience guarantees.