Who Would Win??
Seven limits staged as matchups: the harmonic series diverges, the geometric series sums to 1, (1 + 1/n)ⁿ → e, xˣ → 1, n!/nⁿ → 0, the alternating harmonic series can be rearranged to any sum, and the derivative.
Use this cartoon
Free for classrooms, worksheets, slides and other non-commercial use under CC BY-NC 4.0, with credit to Ben Orlin.
Cartoon by Ben Orlin, Math with Bad Drawings. https://cartoons.mathwithbaddrawings.com/2025-09-10-who-would-win/ (CC BY-NC 4.0)Transcript
[1] 1 + 1/2 + 1/3 + 1/4 + 1/5 + … TEENSY TINY PIECES vs. LOTS AND LOTS OF PIECES. WINNER: lots-and-lots-ness (score: ∞)
[2] 1 + 1/2 + 1/4 + 1/8 + 1/16 + … TEENSY-WEENSY TINY PIECES vs. LOTS AND LOTS OF PIECES. WINNER: teensy-weensy-tiny-ness (score: 1)
[3] lim_{n→∞} (1 + 1/n)ⁿ: TEENY-TINY GROWTH RATE vs. MEGA-LONG TIME IN WHICH TO GROW. WINNER: weirdly enough, e
[4] lim_{x→0⁺} xˣ: SUPER-FAST SHRINKING vs. SUPER-BRIEF TIME IN WHICH TO SHRINK. WINNER: the brevity of time, which makes all of our powers seem like zero
[5] lim_{n→∞} n!/nⁿ: WAYS TO MAKE A PLAYLIST (NO REPEATS) vs. WAYS TO MAKE A PLAYLIST (ALL THE REPEATS YOU WANT). [Playlists: Pink Pony Club, Come on Eileen, Rolling in the Deep / Pink Pony Club ×3, Hey Ya] WINNER: repeats, repeats, repeats
[6] 1 − 1/2 + 1/3 − 1/4 + 1/5 − …: LOTS OF TEENY-TINY REVENUES vs. LOTS OF TEENY-TINY EXPENSES. WINNER: as written, revenues win by a nose, but with a little creative accounting you can cook the books however you want
[7] lim_{h→0} (f(x+h) − f(x))/h: A VERY SHORT DISTANCE vs. A VERY BRIEF TIME. WINNER: Newton and Leibniz (joint victory)