When Can You Switch Limits in Calculus?

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Cantor’s Paradise

24 min readJul 22, 2021

Buckle up. Real analysis pulls no punches.

limit as x approaches a limit as y approaches b of f of x and y equals with a question mark limit as y approaches b limit as x approaches a of f of x and y equals with a question mark limit as (x, y) approaches (a, b) of f of x and y. Basically, does switching the order of limits or taking both at the same time affect the output?

In my previous article, Let’s Derive the Power Rule from Scratch!, I ended the proof by citing the Moore-Osgood theorem to justify switching limits to prove the power rule for irrational powers. While I think it’s fair game to cite basic theorems, I’m a little unhappy with how I pulled out an advanced theorem from real analysis without explanation, as…

When Can You Switch Limits in Calculus?

Written by Joseph Mellor