Asymptote Misconceptions

Can a function cross its horizontal asymptote? Can it be defined on its vertical asymptotes? Most students say no to both questions. But the answer to both questions is actually yes. At the beginning of every year I have to clear up this common misconception. So, let me write it all out so it’s cut and dry. We’ll start by examining a bunch of examples.

Functions that cross their horizontal asymptotes

f(x)=\frac{x}{x^2+1} (also try f(x)=xe^{-x^2})

f(x)=\frac{4x+1}{x^2-2}

f(x)=\frac{\sin{x}}{x}

f(x)=-e^{-2x}+e^{-x}

Functions that are defined on their vertical asymptote

Functions defined on their vertical asymptotes are a bit more contrived, but they are completely legitimate and still pass the “vertical line test.” These functions must be defined piece-wise, like so:

f(x)=\left\{\begin{matrix}&x   &\text{if}\;  x\leq0\\ &1/x &\text{if}\; x> 0\end{matrix}\right.

f(x)=\left\{\begin{matrix}&0   &\text{if}\;  x=0\\ &1/x^2 &\text{if}\; x\neq 0\end{matrix}\right.

What IS an asymptote?

Said loosely, an asymptote is a line that a curve gets closer to as it tends toward infinity (whereby we mean, anywhere on the outskirts of the coordinate plane). Feel free to read the wikipedia entry for all the gory details. It even mentions ‘curvilinear’ asymptotes–asymptotes that aren’t lines. Our Precalculus text mentions them too, but we don’t make a big deal about it.

The most rigorous definition of an asymptote our students see involves limits. And even then, we show you how to evaluate limits but don’t prove limits (with \epsilon\delta proofs!). That’s a bit different than when I took Calculus in high school. I remember learning the proofs. For those who miss them in high school, though, have no fear–a college course in elementary real analysis will feed your hunger! 🙂

So, to my students: I hope this helps clear up some issues you might have with asymptotes and frees your mind from the prototypical examples of asymptotes you might be used to!

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5 thoughts on “Asymptote Misconceptions

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