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 Dec 22 comment Does $(x, y) \to (e^x \cos y, e^x \sin y)$ map open sets to open sets thanks - see my comment to cactus314's answer Dec 22 comment Does $(x, y) \to (e^x \cos y, e^x \sin y)$ map open sets to open sets thanks, I did see this after searching for the question, but this is a question from my real multivariable calculus course (no complex analysis) so I'm sure there's a way to do this without proving that analytic functions map open sets to open sets Dec 22 awarded Student Dec 22 comment Does $(x, y) \to (e^x \cos y, e^x \sin y)$ map open sets to open sets @peterag - yes, I think that works. I am kicking myself for not seeing this. If you write is as an answer I'll accept it Dec 22 comment Does $(x, y) \to (e^x \cos y, e^x \sin y)$ map open sets to open sets This is a basic multivariable calculus course, so I haven't encountered diffeomorphisms yet, but I'll look into it Dec 22 asked Does $(x, y) \to (e^x \cos y, e^x \sin y)$ map open sets to open sets Feb 20 comment What's your favorite proof accessible to a general audience? Another reason I like it - the more biased the coin is, the longer it takes to get one random bit out Sep 30 awarded Explainer Jan 19 answered Mathematically, why does $[ma]\mathrm{d}x = [mv]\mathrm{d}v$? Sep 25 revised Why is integral of $(\tan x)^3$ not $((\sec x)^2)/2 - \ln(\sec x)$? add latex formatting Sep 25 suggested approved edit on Why is integral of $(\tan x)^3$ not $((\sec x)^2)/2 - \ln(\sec x)$? Sep 24 comment L'Hopsital Rule Understanding Sep 24 comment L'Hopsital Rule Understanding That's not true, $f'/g'$ might be undefined while $\lim f'/g'$ exists. Sep 24 comment L'Hopsital Rule Understanding Fair enough. But why doesn't your answer point out this limitation of your proof? I can comment on your answer instead if you want. Sep 24 comment L'Hopsital Rule Understanding I think any proof or proof intuition for l'Hopital's rule must be roughly as complex as mine. Your proof shows that $\lim f/g = f'/g'$, but l'Hopital's rule is that $\lim f/g = \lim f'/g'$. The later is much harder to prove than the former. In fact, I think you should point out this limitation in your answer Sep 24 comment L'Hopsital Rule Understanding Your proof doesn't work for applying l'Hopital's rule twice, your final expression is a ratio of derivatives without a limit in front. Sep 24 comment L'Hopsital Rule Understanding Because if you apply l'Hopital's rule twice, you are eliminating the zeroth and first order terms, and focusing on the quadratic terms of the functions Sep 24 comment L'Hopsital Rule Understanding then you need to talk about quadratic, cubic, quartic...etc approximations. I know the proof of taylor's theorem isn't very simple, but I find the fact that any nice function has a first, second, etc order approximation to be self-evident Sep 24 comment L'Hopsital Rule Understanding I know, this is how I express the intuition that "the first-order term determines the limit" which I think of when using l'Hopital's rule. Sep 24 answered L'Hopsital Rule Understanding