How to find out asymptotes for any algebraic curve?

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Finding asymptotes for general algebraic curves, particularly those not in the rational function form, requires analyzing the behavior of the curve as x and y approach infinity. For the curve defined by the equation y^3 - x^2y + 2y^2 + 4y + x = 0, asymptotes can be determined by neglecting lower-order terms when x and y are large. This approach allows for identifying dominant terms that dictate the curve's behavior at infinity. Unlike rational functions, the process for more complex algebraic curves lacks comprehensive online resources. Understanding these asymptotic behaviors is essential for analyzing the overall shape of such curves.
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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?
 
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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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