When Can You Switch Limits in Calculus?

Buckle up. Real analysis pulls no punches.

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 it goes against the “from scratch” spirit. In this article, we’re going to prove the Moore-Osgood theorem from scratch. In doing so, we’ll end up diving deep into the foundations of calculus and you’ll get to see how the sausage is made in all its gory detail. By the end of this article, you should have a feel for how real analysis works.

Previous Articles

Before going any further, I would recommend checking out my previous articles What is a Limit, Really? and Uniform and Pointwise Convergence. I will be building off the concepts in those articles, which include

• How to read various math symbols.
• The epsilon-delta definition of a limit.
• How to prove a limit using the epsilon-delta definition.
• Using nets, which generalize sequences, to generalize the definition of a limit.
• Pointwise convergence.
• Uniform convergence.
• How to prove uniform convergence.
• Using nets to generalize the two types of convergence.

I apologize for having to refer back to the previous articles, but the only other option would be to copy large chunks of them and paste them here. With that being said, if you understand all of these points, you should be fine just reading this article. If you understand everything but nets, then feel free to skip Uniform and Pointwise Convergence and skip to the section The General Definition for Limits in What is a Limit, Really?.

Real Analysis

If you’ve ever been a student in an English classroom, you may have had the following conversation:

• “Can I go to the bathroom?”
• “I don’t know, CAN you?”
• “MAY I go to the bathroom?”