Why Can A Graph Pass Through Its Horizontal Asymptote?

Question

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 (Link) 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.

Answer 1

An asymptote can pass through a graph. It only has to approach, but never touch the asymptote when it reaches infinity and negative infinity.

Answer 2

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.