You gave no specific questions, I think I should formulate them myself.
1. Alpha decay. Some types of atoms randomly emit alpha particles with the constant probability. For a large collection of such (identical) atoms a half of atoms will decay after some constant time, called half-life. After two half-life periods, 1/4 of the initial amount of atoms will remain, then 1/8 and so on.
So the number of initial atoms of that collection will decrease up to zero.
`lim 1/2^n = 0.`
2. Atmosphere free falling. Any fixed object which falls in atmosphere asymptotically reaches its terminal velocity. This is an example of a finite nonzero limit in real life. The value of this limit depends on the horizontal surface area of an object and some other factors.
2.1. Warming. A frozen object pulled from a refrigerator slowly reaches the ambient temperature. In other words, the limit of an object's temperature is the ambient temperature.
3. (not from real life but a good limit:)
Imagine that we have a large pie and `n` successive guests, and we give each next guest an `1/n` of the remaining part of a pie. Then the part which remains to you will be `(1-1/n)^n.` Imagine that `n` tends to infinity or simply is large (the total number of guests goes up and the one's part goes down correspondingly).
Then the remaining part will be about `1/e` because
Here `e=2.71828...` is the base of natural logarithms.
3.1. Compound interest.
`lim_(n-gtoo) (1+r/n)^n=e^r` occurs when we consider compound interest with the large compounding frequency `n` (`r` is the nominal interest rate).