Graphical Animation to Demonstrate the Spawning of Asymptotes from a Missing Point on the Graph of a Rational Function
Jason Laska


The rational function f(x)=(x²+4x-5)/(x-1) = [(x+5)(x-1)]/(x-1) = x+5, when x¹1 has a removable discontinuity at (1,6). By adding a single controlled parameter "A" to the equation one can use this rational function f(x)=(x²+4Ax-5)/(x-1) to simulate the spawning of asymptotes from the removable discontinuity. In the following example the parameter is being decreased resulting in the following asymptotes:



When the parameter increases it can be seen that the function spawns asymptotes in the opposite direction of the previous animation.



It can therefore be deduced that, in this instance of rational functions, the spawning of asymptotes is directly related to the coefficient of the middle term.