I recently closed out the academic portion of Pre-Cal. We're in the middle of producing a short film and then tackling some small projects that summarize everything we've done this year. The final act of their tested curriculum involved a little bit of calculus. We went through finding limits by direct substitution, determing the value of limits that seem to be undefined, limits at infinity, and derivatives.
Every year I play a dirty trick on them in the section on limits at infinity:
It's an exercise that draws inspiration from this test question by Sam Shah which was in turn inspired by a post from Bowman Dickson. Basically, what happens when you present a student with the unexpected? In the 90 or so papers I have, there are dozens of students that made the same mistake on this question. There were no in class examples like it. Though I did mention that the largest exponent has control of behavior throughout the course of the material. This is the kind of thing that makes it clear who processed the full concept and who was pattern matching to examples.
I would fall prey to these oversights in school all the time. A particularly nasty Thermodynamics exam I took was full of things like this.
Now the conundrum: is this a good test question because it showcases who internalized the concept completely? Or a bad test question because the answer is so obvious to me, the seasoned veteran, and hahaha can you believe these inexperienced teenagers are no match for me?