Javier Badia
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 Jun1 comment Proving that $\lim\limits_{x \to 0}\frac{e^x-1}{x} = 1$ I guess I was looking for answers that use mostly the definition of a limit together with some of $e^x$'s properties, but I'm not sure that's practical. Jun1 comment Proving that $\lim\limits_{x \to 0}\frac{e^x-1}{x} = 1$ I'm seeing that I haven't really thought this through. I would say that the only definition of $e^x$ that makes this problem an interesting problem is the compound interest one, since the ones you suggest are related to the fact that $(e^x)' = e^x$. But then it seems that the problem reduces to showing that the derivative of $\lim_{n \to \infty}(1+\frac{x}{n})^n$ is equal to itself. Jun1 comment Proving that $\lim\limits_{x \to 0}\frac{e^x-1}{x} = 1$ I hadn't really thought about how to define $e^x$, but I guess it would have to be $e^x = \lim_{n\to \infty}(1+\frac{x}{n})^n$, because if you use the other ones that were suggested (power series, differential equation), then the proof becomes almost trivial. I guess that complicates things, doesn't it? Jun1 comment Proving that $\lim\limits_{x \to 0}\frac{e^x-1}{x} = 1$ @Thomas: Isn't using the power series representation of $e^x$ pretty much equivalent to saying that $(e^x)' = e^x$? Jun1 asked Proving that $\lim\limits_{x \to 0}\frac{e^x-1}{x} = 1$ May28 comment How do you explain paradoxes to non-mathematicians? Those are not paradoxes. May26 comment How to show $|\sin(x+iy)|^2=\sin^2x+\sinh^2y$ @Derrick: the idea is that we want to be left only with the circular and hyperbolic sines, so we use the identities I mentioned to replace the cosines with sines. I'm on my iPod so I can't really type out all the TeX, but that is the gist of it. May26 revised How to show $|\sin(x+iy)|^2=\sin^2x+\sinh^2y$ added 372 characters in body May26 answered How to show $|\sin(x+iy)|^2=\sin^2x+\sinh^2y$ May20 accepted Confused about Wikipedia page on differential forms May19 comment Confused about Wikipedia page on differential forms So, just to make sure I understand: Suppose we're working in $\mathbb{R}^2$, and instead of $x,y$ I call them $f_1, f_2$ for clarity, so they're functions from $\mathbb{R}^2$ to $\mathbb{R}$. Then we can say that $f_1(a, b) = a$ and $f_2(a,b)=b$? May17 asked Confused about Wikipedia page on differential forms May13 awarded Fanatic May4 accepted How to prove $\lim_{(x,y)\to(0,0)} \frac{xy}{x+y} = 0$ May4 comment How to prove $\lim_{(x,y)\to(0,0)} \frac{xy}{x+y} = 0$ I guess this is a simpler way to know. How did you think of approaching along $y=x^2-x$? May4 comment How to prove $\lim_{(x,y)\to(0,0)} \frac{xy}{x+y} = 0$ @ArturoMagidin: Oh, I thought you could just sort of ignore it. This means that the limit doesn't exist, then? May4 comment How to prove $\lim_{(x,y)\to(0,0)} \frac{xy}{x+y} = 0$ @Dylan: Well, the function is not even defined along that line. Does that matter? May4 asked How to prove $\lim_{(x,y)\to(0,0)} \frac{xy}{x+y} = 0$ Apr30 comment Can I apply Rolle's theorem in reverse? Thank you for your answer. While it helped me with the problem, David's answered the point of the question, so I accepted his. Thank you anyway. Apr30 accepted Can I apply Rolle's theorem in reverse?