To give a bit of background: I am trying to figure out what amount of substance X to continuously add over a time interval in order to keep it constant in a system where substance X has the half-life $t_{1/2}$.
This is easy as long as I settle on a certain time scale. The amount of X lost in 1h is:
$\Delta X_{60'} = X_0-X_0 \left(\frac{1}{2}\right)^{\frac{60'}{t_{1/2}}} $
In which case adding $X_0-X_0 \left(\frac{1}{2}\right)^{\frac{60'}{t_{1/2}}}$ per hour would keep the system from decaying - but it would not keep the amount of X constant at the starting level within that time interval. For that, one would have to calculate $\varepsilon$, the amount of X lost at the smallest possible time unit ($\delta$) and scale that up to whatever unit of time $T$ one finds feasible for conducting the actual physical experiment:
$ \Delta X_T = \varepsilon \frac{T}{\delta} $
I was thinking, that amount (scaled back up to $T$, say 60') would be given by:
$ \lim_{t\to0} \left( \frac{2^{\frac{t}{t_{1/2}}} X-X}{2^{\frac{t}{t_{1/2}}}} \cdot \frac{T}{t} \right) $
But, to me, this seems to converge to $0$ independently of $T$, or $t_{1/2}$. Have I miscalculated anything? Is there a better solution to this?
The limit is:
$\lim_{t\to0} \left( \frac{2^{\frac{t}{t_{1/2}}} X-X}{2^{\frac{t}{t_{1/2}}}} \cdot \frac{T}{t} \right) = \lim_{t\to0} \left( \left( X-X\left(\frac{1}{2}\right)^\frac{t}{t_{1/2}}\right) \cdot \frac{T}{t}\right)$
$ = XT \lim_{t\to0} \left( \frac{1 - \left(\frac{1}{2}\right)^\frac{t}{t_{1/2}}}{t} \right) = XT\lim_{t\to0} \left( \frac{1 - e^\frac{\ln(1/2) \cdot t }{t_{1/2}}}{t}\right)$$
Letting $ k = \frac{\ln{\frac{1}{2}}}{t_{1/2}}$ and using the fact that $\lim_{n\to0} (1 + n)^{\frac{1}{n}} = e $, then our expression follows as:
$= XT\lim_{t\to0} \left( \frac{1 - e^{kt}}{t}\right) = XT\lim_{t\to0} \left( \frac{1 - (1+t)^{\frac{1}{t} \cdot kt}}{t}\right) = XT\lim_{t\to0} \left( \frac{1 - (1+t)^k}{t}\right)$
By the L'Hôpital's rule (the limit of a quotient of derivable functions is equal to the limit of the quotient of the derivatives of these functions):
$ = XT\lim_{t\to0} \left( \frac{ - k(1+t)^{k-1}}{1}\right) = XT \cdot (-k) = \frac{-XT \ln\frac{1}{2}}{t_{1/2}}$