Server Load Response Latency (Computer Science)

In a distributed cloud microservice architecture, software performance engineers evaluate the average network response latency $T(N)$ (in milliseconds) as a function of the concurrent active compute node count $N$:

As the cluster auto-scales dynamically during peak traffic, the number of worker nodes becomes very large ($N \to \infty$). Systems architects must determine the asymptotic latency ceiling to ensure the application complies with Service Level Agreements (SLAs).

1. Apply Growth Dominance filtering to identify the dominating terms in $T(N)$.
2. Evaluate the limit analytically:
   $$\lim_{N \to \infty} \frac{12N^2 + 45N}{3N^2 + 8}$$
3. State the horizontal asymptote and explain how this value dictates cloud scaling limits.