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 12h comment How to prove that $\lim\limits_{x\to0}\frac{\sin x}x=1$? How do you prove that $y=x$ is that tangent? Feb 26 comment How to prove that a polynomial of degree $n$ has at most $n$ roots? It's obvious that $f=(x-a)q$ if $a$ is a root, but how can that be demonstrated? Feb 7 comment How to derive: $\left(1 + \frac{1}{n}\right)^n < 1 + 1 + \frac{1}{2}+…+\frac{1}{2^{k-1}}$ For what $n$ is this supposed to hold? Feb 7 comment How do I show that the probability of the union of events is not larger than the sum of the individual probabilities? That's a really good idea. Thank you! Feb 7 comment How do I show that the probability of the union of events is not larger than the sum of the individual probabilities? @NobleMushtak Not quite, we are supposed to show an inequality and the $A_n$ are not necessarily disjoint. Dec 4 comment Can we express the roots of all polynomials in terms of roots of some special polynomials? @theREALyumdub It isn't with just ordinary radicals $\root n\of m$, but with the Bring-radical, you always can (as far as I understood). Oct 19 comment Can we express every partial order with these two combinators? @GitGud Yes, precisely. Sep 30 comment How to prove that $\lim\limits_{x\to0}\frac{\sin x}x=1$? @R R It addresses them in the intuitive sense: Imagine reducing one angle until it becomes zero and then continuing to move the line you moved, that's what the construction for negative angles looks like. For this particular application, it's sufficient to show that $\lim_{x\to0^+}{\sin x\over x}=0$ though (if I recall correctly, it has been some time). Aug 24 comment How to prove that $\lim\limits_{x\to0}\frac{\sin x}x=1$? @Asker123 No, because ${\sin 0 \over 0} = {0\over 0}$ and you cannot divide by $0.$ A graphing calculator has finite precision, what tells us that the $0$ it displays isn't actually a $0.000000000000000000012445823?$ Jul 3 comment How to prove that $\lim\limits_{x\to0}\frac{\sin x}x=1$? @goblin The goal is to go from an intuitive understanding of $\sin$ as a geometrical relationship to a function. Using such an implicit definition would be quite dissatisfactory to a student. Apr 17 comment What is the determinant of the sum of a diagonal matrix and a matrix of ones? @HagenvonEitzen When exactly one entry in the diagonal is $1$, the result is $\prod_i(a_{ii}-[a_{ii}\neq1]),$ where $[p]$ is the Iverson bracket. If more than one entry is $1$, the result is $0$. Apr 17 comment What is the determinant of the sum of a diagonal matrix and a matrix of ones? @deinst I think that covers it. Would you mind copying the relevant part of that answer into an answer to this question so I can mark this as accepted? I can do this myself, too, if you don't want to. Apr 11 comment How to prove that $\lim\limits_{x\to0}\frac{\sin x}x=1$? While this is indeed an interesting approach, integrals haven't been taught at the point where this limit is proved. Thank you for your answer though. Mar 23 comment What is the lower bound for an algorithm that reconstructs a permutation? I fixed a couple of typos. I'm sorry if that changes your answer. Mar 13 comment How to prove that $\lim\limits_{x\to0}\frac{\sin x}x=1$? Indeed, it's easy to see that this holds if one uses a series, but this question starts on the prerequisite that one does not use a series. Mar 3 comment Is there a totally ordered set we can map any other totally ordered set to? I edited the question because I thought that the case $|O|=|\mathbb N|$ might be interesting, too. Thank you for researching this! Mar 3 comment Is there a totally ordered set we can map any other totally ordered set to? @AndresCaicedo Thank you. I'll see if my university's library has that. Dec 8 comment How to construct a polynomial from a radix-term? Ah yes, that makes sense. I'm sorry for my confusion. Dec 8 comment How to construct a polynomial from a radix-term? That was a typo, it should have been $\alpha\in\mathbb Q\setminus\{0\}$. I'm a bit confused about how you explain the translation that makes a polynomial $p'$ with $p'(\alpha^{-1})=0$ from a polynomial $p$ with $p(\alpha)=0$. Your explanation does not really enlightens me. Maybe that's just because I am a bit hungover. Dec 8 comment How to construct a polynomial from a radix-term? I'm afraid I might have misunderstood something, but what guarantees that $\alpha\in\mathbb Q\{0\}$? Isn't $\alpha$ the value of an arbitrary algebraic expression with $\alpha\neq0$?