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Vsauce made a video recently on counting past infinity, and he represented the set of natural numbers to infinity with a set of lines, where each successive line is a smaller distance away from the previous line than before, and each line becomes smaller, giving an infinite series in a finite space. Then later I was messing around on Desmos with sine graphs. I put in $$y=\sin(x)\times\sin(x^{-2 })$$ And the graph looks very similar to the way Vsauce represented infinity. I was wondering if this graph crosses the $x$ axis infinitely many times, between the interval $[-1,0]$ or $[0,1]$ and if graphs like this are special, or if I've come across something quite trivial.

(I had to type the equation normally because I'm not sure how to do it any other way on my phone)

Laurent Duval
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Rohan
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3 Answers3

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Since $\sin x$ is zero at $\pi,2\pi,\ldots$, $(\sin x)(\sin x^{-2}) $ is zero at $\dfrac{1}{\sqrt \pi},\dfrac{1}{\sqrt {2\pi}},\ldots $, so the answer to your question is "yes". $\sin\left(\dfrac1x\right)$ is a simpler example with the same property.

Mark S.
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Here's an intuitive way to think of it.

Consider a more general case:$$y=\sin(\frac{1}{x})$$

When $x$ gets closer and closer to $0$, there will be an infinite amount of times it crosses the x-axis.

This is because $\frac{1}{x}$ approaches $-\infty$ from the left and $\infty$ from the right. When $\frac{1}{x}$ gets very close to the y-axis it will grow faster and faster in either the positive or negative direction. This rapid growth will make $\sin(\frac{1}{x})$ oscillate faster and faster as well, and it will cross the x-axis an infinite amount of times as $x$ gets closer to its asymptote, the y-axis because there is essentially infinite growth.

Your equation follows the same idea.

Vasting
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First, this function $f$ can be defined at $0$ by continuity, with value $0$. Now take $x_k = \frac{1}{\sqrt{2\pi k}}$, $k >0$. All $x_k$ are in $[0,1]$, and $f(x_k)=0$, an infinity of times in the interval, and increasingly denser near $0$.

A lot of them are quite easy to build, and motivate of lot of exercices for students to study continuity, differentiability, convergence, etc.

Laurent Duval
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