$\tan$ function at $\pi/2$

Question

$$\tan(\theta) = \dfrac{y}{x}$$ Suppose initially the angle $\theta$ is $0$.
As $\theta$ increases to $\pi/2$, the denominator $x$ decreases and the numerator $y$ increases, so $\tan(\theta)$ becomes arbitrarily large.

Exactly at $\theta=\pi/2$ the function becomes +infinity. Now if we increase $\theta$ even slightly, the denominator becomes $-ve$ and the function value suddenly changes to -infinity.

I'm finding it hard to wrap my head around this. Is there anything in real life that behaves like this? that is, changes its value from high positive to high negative abruptly..

Answer 1

I'm not entirely sure that this is a question about mathematics, but nature does not seem to have any essential discontinuities except (maybe) around a black hole. However, a black hole doesn't have a sign problem as you are having trouble with.

In particular, there don't seem to be any truly infinite amounts in the real world, at least not of anything involving mass or energy in any (even tiny) amounts, since an infinite amount of even tiny things yields a universe-swallowing black hole (again, maybe).

The fact is that infinity is a mathematical concept. And, it happens to be one that you're using slightly incorrectly. You say that at $\theta=\pi / 2$, $\tan \theta$ becomes $+\infty$, but this isn't entirely true. The output of the function is never $+\infty$! The more correct statement is that for values of $\theta$ between $-\pi/2$ and $\pi/2$, as $\theta$ approaches $\pi/2$, $\tan \theta$ grows without bound. But, it never "becomes" $+\infty$.

There is a similar statement on the other side of the discontinuity. For values of $\theta$ between $\pi/2$ and $3\pi/2$, as $\theta$ approaches $\pi/2$, the value of $\tan \theta$ decreases without bound. But again, it is never actually $-\infty$.

Perhaps what might help is the fact that these two vastly different results actually have a very solid wall$-$a boundary$-$between them! The values of $\theta$ which are below $\pi/2$ are separated from the values of $\theta$ which are above $\pi/2$ by the one number which actually is $\pi/2$. At this one value of $\theta$, the expression $\tan \theta$ does not lean towards either side, but is rather totally undefined! That one input is actually a boundary between the two vastly-different results from both sides.

However, this sort of behavior is rife through many different functions. Vertical asymptotes are very common in many functions in mathematics, and many of them experience this behavior of having opposite unbounded behavior immediately on either side of them, just like the tangent function.

Answer 2

Let's start with functions that are easier to think about:

• The function $1/x$ is finite for $x\ne0$, but not for $x=0$. The function approaches $+\infty$ (which I'll hereafter call $\infty$, as is common practice) as $x$ approaches $0$ from the right, but $-\infty$ as $x$ approaches $0$ from the left. This is why you should consider $1/x$ undefined at $x=0$. Every time you catch someone writing $1/0=\infty$, bear this in mind.
• On the other hand, $1/x^2$ approaches $\infty$ as $x$ approaches $0$, no matter which side you come from.

There's nothing paradoxical about either behaviour; some functions are different from others.

It just so happens that the behaviour of $\tan x$ around $\pi/2$ is more like the first example than the second. There's a simple reason for this: if $x$ is close to $\pi/2$, $$\tan x=\frac{1}{\tan\left(\frac{\pi}{2}-x\right)}\approx\frac{-1}{x-\frac{\pi}{2}}.$$In this approximation, the numerator's $-$ sign causes the function to approach $\infty$ when you approach the discontinuity from the left, not the right.

By contrast, $\tan^2x$ approaches $\infty$ as $x$ approaches $\frac{\pi}{2}$, regardless of direction.

You can exhibit the tangent function's behaviour in real life with the right experiment. Suppose you stand halfway between two tall walls, and shine a laser pointer on to one of them. Relative to your wrist, the dot's height is proportional to the tangent of the angle your laser makes with the horizontal. When the angle goes from just above to just below a right angle, or vice versa, you change which wall the dot ends up on (assuming the walls are tall enough to see the dot; if they're not, you'll need to tilt further from the right-angle case). But if you imagine each wall carrying on underground, and the laser pointing in both directions, this is equivalent to saying the other end is now at the bottom of the wall where you previously saw the dot. I suppose the right way to do this experiment is to climb a tower to be halfway between the walls, then use Darth Maul's lightsaber.

Or, if you'll count drawing on paper as "real-life", draw a circle, one of its diameters, and the tangents (the lines, not the function!) to the circle at each end of that diameter. Now rotate another diameter around the circle, but extended to meet the tangents. When the angle this turning diameter makes with the marked one passes a right angle, the ends switch tangents, and each tangent goes from having a point moving "to infinity" in one direction to moving "from infinity" from the other direction.

Even though $1/x$ is conceptually simpler than $\tan x$, it favours more esoteric physical interpretations. For example, as thermodynamic beta passes through $0$, you can get a concept that may confuse at first, called negative temperature.