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 $t=0$.

Throw a stone obliquely. Due to air friction, the trajectory follows a vertical asymptote.

http://www.mathcurve.com/courbes2d/paraboleamortie/paraboleamortie.shtml