What is an Asymptote?

Hey guys I am not being able to understand the Asymptote. Please say me wheather an Asymptote is a line that really meets the curve or not. If it meets its defination is not saying so beacuse its defination is like this "An asymptopte is a line whose perpendicular distance from any point on the curve tends to zero as the point goes to infinty". This defination is clearly suggesting that the line actually doesn't meet the curve. But while deriving the formula to find out the asymptopte of a curve we have solved the equations of a line(asymptopte) and of curve. How can we do it if it really doesn't meet the curve.
Still another thing; if the line meets the curve isn't it a tangent to that curve. Then the equation of asymptote will be same as that of tangent to that curve.
How can I undersatand this all? If you know any good websites that can provide me the detail infornmation about asymptote please mention it.....
 

HallsofIvy

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Hey guys I am not being able to understand the Asymptote. Please say me wheather an Asymptote is a line that really meets the curve or not. If it meets its defination is not saying so beacuse its defination is like this "An asymptopte is a line whose perpendicular distance from any point on the curve tends to zero as the point goes to infinty". This defination is clearly suggesting that the line actually doesn't meet the curve. But while deriving the formula to find out the asymptopte of a curve we have solved the equations of a line(asymptopte) and of curve. How can we do it if it really doesn't meet the curve.
I don't understand what you mean by "solved the equations of a line(asymptopte) and of curve" It might happen that the curve crosses it asymptote at some x value away from where ever the line is asymptotic to the curve but they do not touch at that x. I, certainly, have never seen an example of finding an asymptote by solving the equation of a line and curve simultaneously. Could you give an example of this.

It is sometimes said that an asymptote is "tangent" to a curve "at infinity" but, of course, that is only an analogy. There is no point "at infinity" actually on a curve.

Still another thing; if the line meets the curve isn't it a tangent to that curve. Then the equation of asymptote will be same as that of tangent to that curve.
Yes, a tangent to a curve meets the curve at the point of tangency. An asymptote does NOT and so is not a tangent.

How can I undersatand this all? If you know any good websites that can provide me the detail infornmation about asymptote please mention it.....
Wikipedia is always a good place to start:
http://en.wikipedia.org/wiki/Asymptote
 
One of my mathematics professors described an asymptote as a 'kiss'. What the equation is actually stating is that the curve is forever pointed towards the axis, but will never touch it. It appears to cross the axis in some cases, be tangent in others, and can become indistinguishable from the axis as well.

It is helpful to think relatively by using different yard sticks to measure the distance of the asymptote from the axis. If you use a meter stick the asymptote may appear to touch the axis, however if you use a micrometer stick, it may appear as though it is going to touch if you keep following it.

If you follow it forever, you will find that the angle at which the curve is approaching the axis decreases with distance. It will never be tangent and it will never touch.

Tangent is when the lines are 'always' equal distance from each other, never converging, never diverging.
 

arildno

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Well, due to the Greek origin of the word "asymptote" (non-touching), it is evidently technically correct to say that a curve should never, ever cross its asymptote.

However, one might ask if such a concept is a particularly useful distinguishing criterion.

Instead, the concept of a line that will, in the limit, have the same behaviour as the curve itself is a more general concept, and thus, presumably, more useful.
Several authors use "asymptote" for such a limiting line.

Note that in the latter case, the curve might well criss-cross the limiting line.
 
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asymptotes simplified

It's a simple concept, Try graphing 1/x as x -> 0. We never touch to the y axis (i.e. x =zero). It is an undefined infinity, an asymptote. .
IFF x were to = 0 then, there'd be no argument.
 
A vertical or horizontal asymptote does not intersect the graph as each one exists outside the domain or range.
 

Infrared

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Actually, you can have a function that intersects its horizontal asymptote an infinite number of times. Take [itex] f(x)= \frac{sinx}{x} [/itex]. It has a horizontal asymptote at y=0 (provided that you use the the more liberal definition of asymptote aldrino gave), but also crosses the x-axis an infinite number of times.
 

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