examples of functions with vertical asymptotes in real life
One example would be the gravitational potential energy of a point in relation to a pointwise mass in space. The closer you are to the point, the faster you go.
http://en.wikipedia.org/wiki/Potential_energy#Potential_energy_for_gravitational_forces_between_two_bodies
If you want simpler examples, take any basic equation that implies a linear connection of two quantities, for example:
- $s=v\cdot t$, where $s$ is the distance traveled and $v$ the speed
- $U=R\cdot I$, Ohm's law
- $m=\rho\cdot V$, connecting density, volume and mass.
In each case, you can find some way to explain vertical asymptotes:
- $s=vt$ means that $t=\frac sv$, meaning that the time it takes to travel a certain distance is very large if our speed is very small.
- $U=RI$ means that $I=\frac UR$, so if the resistance is very small, even small values of $U$ will produce a huge current.
- $V=\frac m\rho$, or in other words, if you want one kilogram of something with a very small density, it will take a huge amount of space.
Physics has lots of examples, but it's already the closest field to math. (Plus, in order to observe asymptotic gravity, you'd need a black hole...)
You could use Walmart.
If shoppers arrive nondeterministically at rate $\lambda$ and are served at nondeterministically at rate $\mu$, the average wait time is
$$\frac{1}{\mu − \lambda} − \frac{1}{\mu}$$
As $\lambda$ approaches $\mu$, the average wait time increases to infinity.
This mostly happens around the holidays.
From distance equals rate times time, you get $r=\frac{d}{t}$ For a fixed distance, the less time you take to cover that distance, the faster you go, with a vertical asymptote at $t=0$.