I have seen that the more a particle has a high energy, i.e $E$, the more its lifetime is short, respecting so the uncertainty principle.
But by the definition of this uncertainty principle :
$E\,\Delta t \geq \dfrac{\hbar}{2}$, I can write :
$\Delta t \geq \dfrac{\hbar}{2E}$, then $\Delta t$ has a lower limit and not an upper limit.
If this would be an upper limit, this would mean that $\Delta t$, i.e. the apparition time, should be observed in a time interval lower than $\dfrac{\hbar}{2E}$ : for example, if a detector had a time resolution greater than $\dfrac{\hbar}{2E}$, the particle could not be detected, could it ?
So in which case can we write : $\Delta t \leq \dfrac{\hbar}{2E}$ ??
It seems that I have confusions with this principle. Any clarification is welcome