When $0 \leq t < T$, the limit:
$$ \lim_{n \to \infty} \left( \frac{t}{T} \right)^n = 0 $$
is zero because the ratio $(t/T)<1$, so, when $n$ increases, the term $(t/T)^n$ decreases.
Is there an analitical way to prove that the limit is zero? My concern is that my words are too much qualitative.
Set $\lambda=t/T\in[0,1)$. The function $n\mapsto\lambda^n$ is decreasing and bounded below by $0$. Therefore $\lim_{n\to\infty}\lambda^n=\inf_{n\in\mathbb{N}}\lambda^n$.
Suppose $\inf_{n\in\mathbb{N}}\lambda^n=c>0$. Then there exists $N$ such that
\begin{align*} \lambda^N<(2-\lambda)c. \end{align*}
This means that
\begin{align*} \lambda^{N+1}<\lambda(2-\lambda)c<c \end{align*}
which follows from $(x-1)^2\geq0$ for any $x\in\mathbb{R}$. This contradicts $c$ being the definition of the infimum. Therefore $c=0$ and so $\lim_{n\to\infty}\lambda^n=0$.
