# How to find out asymptotes for any algebraic curve?

• s0ft
In summary, while there are many resources available for finding asymptotes of rational functions, there is a lack of information on how to find asymptotes for general algebraic curves or polynomials. To find asymptotes for a curve in the form f(x,y)=0, where an explicit definition for y is not given, one can examine the behavior of the equation for large values of x and y and neglect certain terms that are numerically insignificant.
s0ft
There are a lot of contents regarding finding vertical, horizontal and oblique asymptotes for the so called 'rational' functions online. All of these curves are given in the form y=f(x)=(g(x))/(h(x)).
But as far as my search results go, there are none regarding general algebraic curves/polynomials.
I'd like to know, for example, how do you find all the asymptotes for the curve:
$y^3 - x^2y + 2y^2 +4y + x = 0$
i.e. for a curve of the form:
$f(x,y) = 0$ where an explicit definition for $y$ has not been provided or would be inconvenient?

Well that one you find them by looking what happens when x and y become very large and you can neglect certain terms as being numerically negligible compared to certain others.

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## 1. What is an asymptote?

An asymptote is a line that a curve approaches but never touches. It can be thought of as a boundary for the curve.

## 2. How do I determine if a curve has an asymptote?

To determine if a curve has an asymptote, you can look at the behavior of the curve as the x and y values approach infinity. If the curve approaches a specific line or value, then it has an asymptote.

## 3. Can all algebraic curves have asymptotes?

No, not all algebraic curves have asymptotes. Some curves may have infinite or non-existent asymptotes, while others may not have any at all.

## 4. How do I find the equation of an asymptote?

The equation of an asymptote can be found by setting the limit of the curve as x or y approaches infinity equal to the equation of the asymptote. This can be done by using algebraic techniques such as dividing polynomials or taking the derivative of the curve.

## 5. Can I have more than one asymptote for a single curve?

Yes, it is possible for a curve to have more than one asymptote. This can happen if the curve approaches different lines or values as x or y approaches infinity.

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