So far, so good.
Now and (This is what I assume you mean by 3 x 105). I think there may be a problem here: if the initial rate of growth is 300,000 cells per month, the terminal value will be reached very soon. So to double from 106 to 212 in 6 months doesn't seem very sensible. But based on this value...
(using a spreadsheet to do the number-crunching)
This gives the limiting value as 212, which is reached almost immediately. So, have we got the numbers wrong?
PS. Of course! Perhaps you mean . That makes a lot more sense. This gives , , and the limiting value as .