But now we can’t teach proofs to these students! The method is rather more complicated when we’ve got more than one variable to work with, not least because of the more complicated distance formula to work with. What used to happen was that students would have developed some facility with proofs back in first and second semester calculus, which could then be brought to bear on this new situation. But now they have no background and cannot, in general, absorb both the logical details of challenge-response proofs and the complications of multiple variables at the same time. And so we show them a few jury-rigged tricks and assure them that within the rest of the course they won’t have to worry about it. I’d almost rather dispense with limits entirely than present this Frankenstein’s monstrosity.
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