How to find out asymptotes for any algebraic curve?

  • Context: Undergrad 
  • Thread starter Thread starter s0ft
  • Start date Start date
  • Tags Tags
    Curve
Click For Summary
SUMMARY

This discussion focuses on finding asymptotes for general algebraic curves, specifically those defined by implicit functions such as the curve given by the equation y^3 - x^2y + 2y^2 + 4y + x = 0. Unlike rational functions, which have well-documented methods for determining vertical, horizontal, and oblique asymptotes, general algebraic curves require an analysis of the behavior of the function as x and y approach infinity. The key technique involves neglecting lower-order terms to identify dominant behavior in the curve.

PREREQUISITES
  • Understanding of algebraic curves and polynomials
  • Familiarity with asymptotic analysis
  • Knowledge of implicit functions and their properties
  • Basic calculus concepts, particularly limits
NEXT STEPS
  • Study asymptotic behavior of implicit functions
  • Learn techniques for analyzing polynomial curves
  • Explore methods for finding asymptotes in higher-dimensional algebraic curves
  • Investigate the use of software tools for curve analysis, such as Mathematica or MATLAB
USEFUL FOR

Mathematicians, algebra students, and anyone interested in advanced calculus or curve analysis will benefit from this discussion.

s0ft
Messages
83
Reaction score
0
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?
 
Physics news on Phys.org
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.
 
Last edited:

Similar threads

Undergrad Asymptotes of Rational Functions....
  • · Replies 3 ·
Replies
3
Views
2K
Determining the horizontal asymptote
  • · Replies 6 ·
Replies
6
Views
2K
Undergrad How to evaluate the enclosed area of this implicit curve?
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad Find the area enclosed by the curve y = x csch(x+y)
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Questions about algebraic curves and homogeneous polynomial equations
  • · Replies 4 ·
Replies
4
Views
3K
Graphical Transformations and Finding the Equation of a Curve
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Finding the sum of heights under a curve
  • · Replies 3 ·
Replies
3
Views
1K
Undergrad Understanding Mixed Partial Derivatives: How Do You Solve Them?
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Understanding Theorem 13 from Calculus 7th ed, R. Adams, C. Essex, 4.10
  • · Replies 10 ·
Replies
10
Views
3K
MHB Need help plotting a curve, no calculus knowledge....
  • · Replies 1 ·
Replies
1
Views
2K