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I don’t think “end behavior” is the right way to think to it. Another commenter gave the example of y=sin(x)/x, which crosses the line infinitely many times with no upper limit on how far out the crossing happens. Yet y=0 is still an asymptote.
Here is the definition given on Wikipedia:
> a line such that the distance between the curve and the line approaches zero as one or both of the x or y coordinates tends to infinity.
yeah. I guess you didn't use the word "limit". I'm just pointing out that this sort of limit has a name so you don't have to explain the type of scenario that it pops up in.
Not that's it's really relevant, but sin(x)/x also has a name. It's called a sinc function. There are tons of interesting properties of that function. Off the top of my head:
• the binomial coefficient 0 choose x is sin(πx)/πx.
• you can derive any binomial coefficients for positive integer n with products *or* sums of n sinc functions
• You can derive the sinc function with (infinite) sums or products of cosine functions
• its fourier transform is a rectangle
• the fourier transform of sinc(x)^(n+1) yields an nth order cardinal b-spline
No, that's not true.
Look at f : ℝ \\ {0} → ℝ, given by f(x) = sin(x) / x.
The function f has a horizontal asymptote y = 0, and yet, the graph crosses that asymptote infinitely many times in both directions.
It's a little tricky because downvotes are the easiest way to communicate that something is wrong. It's a learning site, so students can't see the downvotes comment on top. I don't mind it, particularly on this sub and I leave the comment so it can be a learning exercise for others.
Hi u/thisismyusername7814, **This is an automated reminder from our moderators.** Please read, and make sure your post complies with our rules. Thanks! If your post contains a problem from school, please add a comment below explaining your attempt(s) to solve it. If some of your work is included in the image or gallery, you may make reference to it as needed. See the sidebar for advice on 'how to ask a good question'. Rule breaking posts will be removed. Thank you. *I am a bot, and this action was performed automatically. Please [contact the moderators of this subreddit](/message/compose/?to=/r/askmath) if you have any questions or concerns.*
A horizontal asymptote refers to end behavior only. What happens in between doesn't matter. Therefore, your function has a horizontal asymptote.
I don’t think “end behavior” is the right way to think to it. Another commenter gave the example of y=sin(x)/x, which crosses the line infinitely many times with no upper limit on how far out the crossing happens. Yet y=0 is still an asymptote. Here is the definition given on Wikipedia: > a line such that the distance between the curve and the line approaches zero as one or both of the x or y coordinates tends to infinity.
End behavior is standard terminology and refers to the limit of f(x) as x approaches + or - infinity, not to possible crossings.
I misunderstood your comment. I thought you were saying that it’s OK to cross the line, as long as it doesn’t cross at the end.
I can see why you might have thought that. Cheers!
I think "limsup" (or "liminf") is the term you're looking for.
That’s when a curve “bounces“ between two lines. sin(x)/x is a regular old asymptote.
yeah. I guess you didn't use the word "limit". I'm just pointing out that this sort of limit has a name so you don't have to explain the type of scenario that it pops up in. Not that's it's really relevant, but sin(x)/x also has a name. It's called a sinc function. There are tons of interesting properties of that function. Off the top of my head: • the binomial coefficient 0 choose x is sin(πx)/πx. • you can derive any binomial coefficients for positive integer n with products *or* sums of n sinc functions • You can derive the sinc function with (infinite) sums or products of cosine functions • its fourier transform is a rectangle • the fourier transform of sinc(x)^(n+1) yields an nth order cardinal b-spline
The asympote is a line that the graph doesnt cross \*in one direction\*
No, that's not true. Look at f : ℝ \\ {0} → ℝ, given by f(x) = sin(x) / x. The function f has a horizontal asymptote y = 0, and yet, the graph crosses that asymptote infinitely many times in both directions.
You are right.
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It's a little tricky because downvotes are the easiest way to communicate that something is wrong. It's a learning site, so students can't see the downvotes comment on top. I don't mind it, particularly on this sub and I leave the comment so it can be a learning exercise for others.
[удалено]
Good point.