Consider a rectangle with vertices $2+iT$, $2-iT$, $-1+iT$ and $-1-iT$ where $T>3$ is not an ordinate of zero of Riemann zeta function $\zeta(s)$. Then if $N(T)$ is the number of zeros of the Riemann zeta function upto height $T$, then in The Theory of Hardy's Z function by Aleksandar Ivic pg. 10-11 we have $$N(T)=\frac{T}{2\pi}\log\left(\frac{T}{2\pi}\right)-\frac{T}{2\pi}+\frac{7}{8}+\mathcal{O}(\log T)$$
So we have $$\lim_{T\to \infty}\frac{N(T)}{ \frac{T}{2\pi}\log\left(\frac{T}{2\pi}\right)-\frac{T}{2\pi} }=1 $$
Question: How can we take the limit as $T$ goes to $\infty$? Is $T$ arbitrary? If yes then how can we consider a rectangle with arbitrary height?
T is not COMPLETELY "arbitrary" as we require that "T> 3" but other than that, yes, T is a variable and can take any value greater than 3.