Asymptotes

[apple2357]

My basic understanding of an asymptote is a line that the curve approaches as it gets very large ( either x or y values).
So the usual examples like y=1/x make a lot of sense.
I have seen this curve on wiki

Asymptote - Wikipedia

en.wikipedia.org

I don't quite understand how this is an asymptote given the curve crosses over the asymptote? Puzzled?

 

[lev888]

My basic understanding of an asymptote is a line that the curve approaches as it gets very large ( either x or y values).
So the usual examples like y=1/x make a lot of sense.
I have seen this curve on wiki

Asymptote - Wikipedia

en.wikipedia.org

I don't quite understand how this is an asymptote given the curve crosses over the asymptote? Puzzled?

Just read the article: "Some sources include the requirement that the curve may not cross the line infinitely often, but this is unusual for modern authors."

What's important is the definition (Oblique case): https://en.wikipedia.org/wiki/Asymptote#Oblique_asymptotes
The limit of the difference between the values of the curve and the line at point x is 0 as x tends to plus/minus infinity. This works regardless of the number of intersections.

 

[apple2357]

Thanks. I must have missed that bit!
Didn’t realise it was a decision.

 

[Dr.Peterson]

The main thing to know is that everyone agrees that a curve can cross its asymptotes (a finite number of times, at least); the only uncertainty would be whether it can happen infinitely, as in that particular graph.

Many students hear that a function "approaches an asymptote but doesn't reach it", and think that means it can never cross. This is reinforced when all the pictures they see are of graphs that don't cross their asymptotes, in order to keep things simple. But it's a false impression, contrary to the definition.

Everything in the definition applies only in terms of "end behavior" -- it doesn't care about "middle behavior".

 

[apple2357]

That's really interesting. The example above is clearly an unusual one. Are there simpler examples of curves that cross asymptotes? Perhaps a finite number of times?

 

[apple2357]

Ah maybe a rational function where the curve might cross a horizontal asymptote! I will come up with one!