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While graphing equations I come across, or find interesting, I found the relationship between graphs $$y=ex^2 \sin\left(\frac{1}{x}\right) \qquad y=ex^2 \qquad y=-ex^2$$

https://www.desmos.com/calculator/nhf6tphhbb

How are these three graphs related, and is there some pattern to their points of intersection?

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    $\begingroup$ It's nothing deep, it's just that the factor $\sin(1/x)$ oscillates between $-1$ and $+1$. $\endgroup$ Nov 5 '19 at 16:45
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As $-1\le\sin{(\frac{1}{x})}\le1$, we have $-ex^2\le ex^2\sin{(\frac{1}{x})}\le ex^2$.

So, the functions $ex^2$ and $-ex^2$ serve as upper bounds for $ex^2\sin(\frac{1}{x})$

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  • $\begingroup$ Goodness, I really am forgetting the basic high school maths! thank you for explaining something which should have been very simple! $\endgroup$
    – Eva
    Nov 5 '19 at 16:49

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