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I am confused why a "horizontal asymptote" that has been passed through by a graph can still be considered an asymptote? Can someone please explain this to me?

I've read another website (http://mathforum.org/library/drmath/view/69843.html) that states "It is a common misconception that it can't EVER touch; the correct idea is that although it approaches the asymptote closer and closer as you move out along the curve, it never actually reaches the asymptote and STAYS there."

But then, in such a situation, what can be considered staying there? If staying there means just one point touching it, then can't the $x$-axis be considered the asymptote of $y=x^2$?

And also, what would constitute approaching the asymptote? How close does the line need to get to the asymptote for it to be considered approaching?

And lastly, if a line in a graph gets very close to an "asymptote" on one side of the "asymptote", then veers completely away from the "asymptote" after passing through it, can this "asymptote" still be considered an asymptote?

Can you please explain this at the level of a high school student who still hasn't learned Calculus? Thank you so much.

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  • $\begingroup$ Read this: en.wikipedia.org/wiki/Asymptote . Acoordingly you are allowed to require that there is an $R>0$ such that the curve and the asymptote don't intersect in points $(x,y)$ with $x^2+y^2>R^2$. $\endgroup$ – Christian Blatter Jul 21 '18 at 9:18
  • $\begingroup$ While the etymology of the word asymptote is a [not]+ syn [together]+ ptotos [falling] is that they shouldn't fall together, that condition is not that useful. So, asymptote gets defined as getting arbitrarily close together. Probably the modern concept should be called symptote. $\endgroup$ – user577471 Jul 21 '18 at 9:28
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An asymptote is a straight line the behaves like the function $f$ that we're working with. Nothing in this idea goes aginst the possibility that the asymptote and the graph of $f$ intersect. Think of $f(x)=\frac{\sin x}x$ and the line $y=0$, for instance. The line is an asymptote of $f$ and the graph of $f$ and the line intersect infinitely often.

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  • $\begingroup$ Carlos Santos My question is though, how close do you need to get to the asymptote to be considered approaching it? And when will a graph be too far away from a line for that line to not be considered it's asymptote. Finally, when can a graph be considered "staying" on a line, such that that line can't be considered an asymptote? $\endgroup$ – Ethan Chan Jul 21 '18 at 9:24
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    $\begingroup$ @EthanChan We say that the line $y=k$ is an horizontal asymptote of the graph of $f$ when (and only when) $\lim_{x\to+\infty}f(x)=k$. This applies to my example. $\endgroup$ – José Carlos Santos Jul 21 '18 at 9:26
  • $\begingroup$ Carlos Santos Wait can you explain to me what limx→+∞f(x)=k means? I've seen this notation before but I don't know what it means, and why it answers my question. Thanks. $\endgroup$ – Ethan Chan Jul 21 '18 at 9:29
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    $\begingroup$ The assertion $\lim_{x\to+\infty}f(x)=k$ means that for every number $\varepsilon>0$, there is a number $M\in\mathbb{R}^+$ such that $x>M\implies|f(x)-k|<\varepsilon$. This answers your questions because it explains the meaning of “horizontal asymptote”. $\endgroup$ – José Carlos Santos Jul 21 '18 at 9:31
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    $\begingroup$ It means that the distance from $f(x)$ to $k$ becomes as small as you wish, as $x$ goes to $+\infty$. $\endgroup$ – José Carlos Santos Jul 21 '18 at 10:04
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An asymptote can pass through a graph. It only has to approach, but never touch the asymptote when it reaches infinity and negative infinity.

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  • $\begingroup$ Okay, but then what counts as approaching the asymptote then? $\endgroup$ – Ethan Chan Jul 21 '18 at 9:19
  • $\begingroup$ When the graph gets nearer and nearer to the asymptote but never touches it. $\endgroup$ – Alan Zhou Jul 21 '18 at 9:22
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    $\begingroup$ The "never touch the asymptote", even though part of the etymology of the word asymptote, is not part of the definition anymore. It is not that useful. $\endgroup$ – user577471 Jul 21 '18 at 9:22
  • $\begingroup$ This is plain wrong. (And a fine example of this common misconception.) Look at the already mentioned example of $f(x)=\frac{\sin x}{x}$. $\endgroup$ – zipirovich Jul 21 '18 at 16:44

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