Asymptotes of Functions
Asymptotes to those that are just learning them seem to just be a set of rules but if you delve deeper you will see
that contrary to popular belief they actually do make sense. It is just often the case that we simply
teach students the rules because we do not believe or want to make the effort or I'm not sure of all teachers reasons
to help students discover why they make sense. The why is taught in great detail in calculus one but I do not believe
the great details are necessary for our students to have a greater understanding than just memorizing the rules. I'm hesitate
to give links on this page because most of them just list the rules, but I invite you first to look at and play with this
geogebra applet that I have developed to aid in the discovery of the asymptotes.
Real-Life Applications & Significance
Anything that has a real-life application is significant because it means we are using it everyday to discover. In the case of asymptotes it is
often true that scientist are trying to find ways to break old asymptotes that are holding them back. For example, in designing air planes an asymptote
for speed of an airplane use to be the speed of sound. Scientist were not able to develop a plane faster than the speed of sound because they did not have materials
to withstand the shockwaves at such a high speed. Once the materials were developed that was no longer an asymptote for the scientist. Other sorts of real life examples
would be a hot cocoa cooling to room temperature as it is left out on the counter, the asymptote would be the temperature of the room or a common example used in mathematics
courses is the decline of medicine such as aspirin in your system. Once you take the aspirin the amount of medicine in your blood stream will go up but
as time goes on the amount in your blood stream will get smaller and smaller and as time goes to infinity then the amount of aspirin will go to zero. Therefore,
our asymptote in this case would be that the amount of aspirin in our blood stream could not be negative or in other words could not go past zero.
Two more humorous examples of asymptotes would be my two pictures located at the top of this page. One is an indication of flappy bird and for those of you who have
played flappy bird you know it is a true statement that you can not pass the line to the green pipes or you will die. Secondly, for those of you who are Lord of the Ring
fans will see the Gandalf graph and understand that Gandalf is saying exactly what an asymptote says, it is a line the function can not pass with exceptions in the horizontal asymptote
world but Gandalf is all about the vertical asymptotes in this case.
Most of the situations above are describing horizontal asymptotes but for a moment let's focus directly on a real life example of vertical asymptotes. I would like to reference
this example from a source I found because I believe it was explained in a wonderful way:
'Vertical Asymptotes also somtimes symbolize quantities that are nonexistent. For instance, if
a preschool wants to build an enclosed playground for its students, it would need a fence. the builders
would consider the best way to lay out their available fencing so that the students have a large
amount of play space. A function to describe the dimensions of a rectangular play space are
w=200/l, where w is the unknown width of the field, l is the unknown length of the field, and
200 square feet is the desired area of the play space. In this case, l=0 is a vertical asymptote,
because not only is it impossible to divide by zero, but it is impossible to have a rectangular
play space that does not have any length!' [McConnel pg. 9]
Links to Videos & Websites
Khan Academy-Visually Identifying Vertical Asymptotes
Purple Math-Vertical Asymptotes
Purple Math-Horizontal & Slant Asymptotes
Cool Math- Horizontal Asymptotes