in which i was almost a mathematical zhuangzi (莊子).

when i woke up this morning, i thought i had dreamt that i wrote a calculus lecture. panicked, i sped to my bookbag and checked my notebook.

luckily, i had actually wrote it,
not dreamt it.

on the other hand, i had hurriedly wrote an analysis lecture. (probably my students could tell.) the topic concerned οne-sided limits, which never appealed to me [1] and i was hard pressed to cover the material in an engaging way. it just seemed dry stuff.

one of these days i have to stop being so indulgent.

the point is that the students learn the standard canon of 1-variable analysιs, not what i find interesting! this is the mathematical equivalent of eating enough vegetables!

so i attempted a compromise:

to show that e^1/x has no limit as x → 0⁺, one needs the "calculus" fact that x ≤ e^x for positive values of x.

as it happens, though, if you define e^x conveniently, then you don't need calculus. so i did.

maybe it was too much, though, or maybe the lecture was already so erratic that it didn't matter.

[sighs]

[1] if one works not on the real line but in space, then a directional limit may be occasionally relevant .. but i don't recall them being that useful.