# examples of functions with vertical asymptotes in real life

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.

• Google "ladder sliding down a wall". Nice function there with square root and two vertical asymptotes. Commented Jan 21, 2015 at 11:33
• nice, but they don't know anything about derivatives (yet). Commented Jan 21, 2015 at 11:35
• You don't need the derivatives stuff, only to derive the form of the function and to point out its two vertical asymptotes and the extreme poins of the definition domain. The derivatives stuff is nice, but you can keep it in the drawer for the time being. Commented Jan 21, 2015 at 11:39
• Could you please turn the comment into an answer with the details? I'm not sure to understand. Commented Jan 21, 2015 at 11:42
• $\tan x$ is no good? what about the rectangular hyperbola $y = 1/x?$
– abel
Commented Jan 21, 2015 at 11:56

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.
• good point, but it may be too "abstract" for them :) (yes, they are really basic students) Commented Jan 21, 2015 at 11:34
• @marcotrevi I added more simple examples.
– 5xum
Commented Jan 21, 2015 at 11:54
• these are all great, i was looking for something "creative", where one could also excercise finding zeroes etc. Here the asymptotes are all in $x=0$... Commented Jan 21, 2015 at 11:59
• @marcotrevi I see, I thought that you only need an example in which to explain the nature of vertical asymptotes in real life.
– 5xum
Commented Jan 21, 2015 at 12:11
• actually these examples help a lot, as I can show them how a proportionality law can give rise to asymptotic behavior. Commented Jan 21, 2015 at 12:59

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

Newton's inverse square law of gravity! Can't much closer to real-life, everyday experiences than gravity!

$F = G\frac {m_1m_2} {d^2}$

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.

• thank you for the hint. By the way, your username reminds me of a Susskind lecture on fish swimming in a black hole's event horizon...and your answer is about gravity. Quantum coincidences? :) Commented Jan 26, 2015 at 11:13

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?

• The coin example is a great decoy for $1/n$! :) thanks! Commented Jan 22, 2015 at 8:02

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.

Slamming the brakes or even crashing in a car?

• Gently -> feel it a tiny bit
• Very hard -> get jerked around

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."

• mm..this does not look "mathy" enough for us :) Commented Jan 21, 2015 at 17:04
• It doesn't go to infinity... if it did, every car crash would destroy the universe. The force reaches a threshold where it kills people, but that force is definitely finite and actually pretty small in the grand scheme of things. Commented Jan 21, 2015 at 19:42
• @ApproachingDarknessFish firstly, no, because no car crash happens in literally zero seconds. Commented Jan 21, 2015 at 19:52

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)

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.