As a math teacher, I tend to get the class involved by finding real-life applications of the math- with functions and vertical asymptotes I am having trouble finding simple enough (rational) functions that describe real-life phenomena. Any help?
ADDENDUM: the only example I could think of is the surface area of a square-based box of fixed volume $V$, i.e. $(4V+2x^3)/x$, where $x$ is the side of the base.
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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 $t=0$.
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Throw a stone obliquely. Due to air friction, the trajectory follows a vertical asymptote.
http://www.mathcurve.com/courbes2d/paraboleamortie/paraboleamortie.shtml
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Slamming the brakes or even crashing in a car?
I'm not thinking too hard about the mathematical model but it's exactly the right concept in that as time goes to zero, physical damage explodes "infinitely."
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There is an example from physics. The tension in a rope hung between two trees.
http://web.mit.edu/8.01t/www/materials/InClass/IC_Sol_W03D1-1.pdf
In order for the rope to be absolutely flat, the tension at the ends must be infinite.
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Newton's inverse square law of gravity! Can't much closer to real-life, everyday experiences than gravity!
The law reads:
$F = G\frac {m_1m_2} {d^2}$
Where $F$ is the gravitational force between two bodies, $G$ is the universal gravitational constant, $m_1$ and $m_2$ are the masses of the two bodies, and $d$ is the distance between the two bodies.
Graphing $F$ as a function of $d$ produces a vertical asymptote at $d=0$. Intuitively, gravity gets stronger as two bodies get closer, but what is the gravity of two bodies in the exact same location? It doesn't make sense.
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Have you tried to describe, how high is the aiming point at the wall in front of you when you rise a rifle at a given angle? (answer: $\mathrm{height} = \mathrm{distance} \times \tan(\text{angle})$ with a vertical ;) asymptote at $\tfrac \pi 2$)
Another 'tan' disguise: how far away from the Earth you need to be to see half of it?
Similar: $\sec x =\tfrac 1{\cos x}$
Not much 'real life' examples, however as much real, as the whole mathematics, I think...
An 'unreal life' example: put a coin into your pocket half past eleven, then 20 to twelve, 10 15 to twelve and so on, at each $\tfrac 1 n$ of an hour to a noon (for $n\in\Bbb N$). How many coins do you have at noon?
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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.
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This example isn't too great if you're looking for a smooth, simple function (since it's an oscillating, discrete one), but I believe it conveys the 'physical idea' of vertical asymptotic behavior nicely, though it's doesn't actually involve one at all.
Imagine a lamp that is on for 1s, then off for 0.5s, then on for 0.25s, then off for 0.125s, and so on; the lamp is switching itself on and off with halfing periods each time. You could ask your students what the state of the lamp is at t = 2 s; it's obviously undefined there.
(I've drawn lines for the sake of identifying how the discrete points bunch closer and closer, to to give 'both states at t = 2' if you allow that idea)
This is of course very different to a continuous function (though of course you can easily interpolate a smooth oscillating function) having a vertical asymptote which tends toward some infinity, but to me, it really conveys that the function isn't just "too big" at the point, but that it's really outside the domain of definition (this example doesn't approach infinity like at a vertical asymptote however).
Not strictly relevant or what you asked for, but carries a very powerful physical interpretation in my opinion (this lamp is impossible, else all time beyond 2s from its activation does not exist)
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any y=f(x) function that divides by (x) has an asymptote, where x=0. Its asymptote is easily offset by (x)<-(x-a) or mirrored with (a)<-(a*sign(a)) This is a simple start. vertical asymptotes are much simpler cases than non vertical ones, where x is also the dividend. Except if your function is a (iterative) logarithm or like all the divergent infinite sums/series. often knowing which factors or exponents grows the fastest (or slowest) can be a bit trickier.
physics is full of reciprocal relationships where ever it simplifies reality just a little bit too much.
calculating intersections of parallel lines also ends up on asymptotic cases, where you divide by 0, if you do not catch the fact that parallel topology has no intersection solutions.
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:
In each case, you can find some way to explain vertical asymptotes: