I'm reviewing some early calculus again, like limits, for doing more stuff with step and impulse functions. And I'm wondering if there's anything wrong with doing it this way, since I don't remember people doing it this way, and neither my calculus books or pdfs do it directly. And I lost my real analysis book years ago.
So for finite limits anyways, since the definition uses the assumed limit L, and we know x-->c , so we know c.
So I'm been just solving some directly plugging x=c+delta, into the |f(x)-L| < e, and it works so far.
https://www.ocf.berkeley.edu/~yosenl/math/epsilon-delta.pdfSo in problem 4 on the bottom of page 3, you would get the polynomial | (delta)^2 +5delta | < e
Which solves for any chosen positive e, you'll get 1 good delta answer.
And for positive delta and epsilon, the pdf works out the same thing basically, and says
|x-2|=delta < (e/6) ==> 6delta < e (or equal)
so with the substitution way , so long as delta is between [0,1], I have
(delta)^2 +5delta < 6delta (or equal) and that's always true. And it worked for the negative delta aswell.
And thats way shorter that counting/ordering number line way
For something like 1/x, I'd say treat L as finite , L=1/delta and try induction type steps and say it goes to infinity. But I was only going to review improper integrals.
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