Limits of Desmos

First off, let me begin by assuring you that the title of this post is not a pun.

In one of my calculus classes recently, we were talking about L’Hopital’s Rule when the following problem came up:

$$\lim_{x\to 0^+} (1+\sin(4x))^{\cot(x)}$$

This limit can be computed with a rather straightforward, although perhaps slightly messy, application of L’Hopital’s Rule, which gives a value of $e^4\approx 54.598$. One of my students was exploring the limit numerically, though, by having Desmos evaluate the function at small positive inputs. This was working well for him for a while until his inputs got so small that Desmos lost it (around $10^{-17}$).

About a week later, we were studying parametric curves when a student came in asking why her graph of $$\begin{cases}x(t) = \sin(t)\\y(t)=\sin(4t)\end{cases}$$ was wrong. I looked at it, though, and told her it wasn’t. Confused, she told me that she had checked her answer in Desmos. Generally trusting Desmos, I called her out and demanded to see her graph. Sure enough, she had graphed the correct function, but she entered the domain as $t\in [0,1000]$. Mathematically this makes no difference to the graph, but it makes a big difference to the graph-drawing software. Her graph looked like this

Parametric Curve with Bad Domain

instead of this, which I got by setting the domain to $t\in [0,2\pi]$ to cover one full period of the curve:

Parametric Curve with Good Domain

Luckily, in both of these cases, I got to have interesting conversations with my students about the software works. All of this is just to show that technology can be great for getting a sense of what is going on in a problem, but it should not be trusted as the source of absolute truth that students sometimes give it credit for.