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When I was drawing some interesting graph of functions, I found a strange phenomenon. A function $f(x)=\tan(\sin x)-\sin(\tan x)$ is well defined on $\mathbb{R}-\left\{\left(\frac{1}{2}+n\right)\pi:n\in \mathbb{Z}\right\}$, and it has oscillation near $x=\left(\frac{1}{2}+n\right)\pi$ by the $\sin(\tan x)$ term. It doesn't oscillate near $0$.

And since $$\lim_{x\rightarrow0}\frac{\tan(\sin x)-\sin(\tan x)}{x^4}=0$$ by taylor expansion, a function defined by $$g(x)=\begin{cases}\frac{\tan(\sin x)-\sin(\tan x)}{x^4}\quad&\mathrm{if}\quad x\neq 0,\, \left(\frac{1}{2}+n\right)\pi\\0&\mathrm{if}\quad x=0\end{cases}$$

is continuous in it's domain. I guessed that $g$ may oscillate at $x=\left(\frac{1}{2}+n\right)\pi$ and not at $x=0$. However, when I plot the graph of $g$ using Desmos or GeoGebra, there's a strange oscillation near $x=0$. Here is the graph.

graph of the function

The similar phenomenon occurs for $x^5$, $x^6$ instead of $x^4$, and I can't understand why it occurs. I tried to investigate sign of $g'$ and get a supremum of $g$ near $0$, but both didn't give me useful information because of complicated calculation. Why this strange oscillation occurs near $0$? Or it's just a numerical error? Any ideas will be appreciated.

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    $\begingroup$ That must be an issue with the softwares since $$g(x) = \frac{1}{{30}}x^3 + \frac{{29}}{{756}}x^5 + \cdots $$ near $x=0$. I used another software to plot $g(x)$ and it looks perfectly fine for $|x|<\frac{1}{2}$ say. $\endgroup$
    – Gary
    Nov 1, 2021 at 4:53
  • $\begingroup$ Which software could plot this well? I want to see the graph. Both Desmos and GeoGebra doesn't work. $\endgroup$
    – sungwpark
    Nov 1, 2021 at 5:00

2 Answers 2

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This is underflow. You are taking the difference between two functions and dividing by $x^4\approx 10^{-16}$. If either of your first numbers has been rounded off by $10^{-20}$, that will cause the error around $0.0001$ that you saw.
Floating point numbers almost always have tiny errors because they are approximated by nearby numbers whose denominator can only be a power of $2$. These errors don't usually matter, down in the 20th decimal place. You have managed to make the errors significant.

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What software/hardware are you using? The problem is when you are doing a "small/small" type of computation on computers, there tend to be a lot of error, as the floating point arithmetic is never precise. In fact, a large part of numerical analysis is about how to handle this problem.

If you use a math software that's more symbolic in nature, you'll be able to see a smooth curve.

E.g., if you type "plot((tan(sin(x))-sin(tan(x)))/(x^4), (x,-1,1))" to https://sagecell.sagemath.org/ and hit evaluate, you'll see Plot

I guess Mathematica will do the same, but I don't have a license for it now.

But if you try to zoom in, try "plot((tan(sin(x))-sin(tan(x)))/(x^4), (x,-0.01,0.01))", you'll see a similar pattern (which really shows the limit of numerical computation):

Plot2

And WolframAlpha isn't doing any better.

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