CPT As a scientist, i implore you to study sinusoidial functions, whether they are cosidered for the univese as it is, just as aether, dissmidmissed for 90 odd years, and misplaced our own jugdgement on physics itself.
The original questions have nothing to do with physics. It's like someone asking "What is 2+3?" and you replying "Let pi = \(\frac{d}{dx}( f(x)+\phi ) = \lim_{g \to a} k\) where k is the field strength and so the answer is \(f\left( x+ \frac{1}{\pi^{e}} \right)\) kilograms" You answer complete gibberish, it's not even coherent. How is + or - a period? Where did anyone mention physical units? You say "I implore you to study sinusoidal functions" but I guarentee you cannot do them yourself. If you could you'd know your answer is completely inappropriate. Fair enough if you're going to post a thread yourself, it's BS from it's inception. But don't post your mindless tosh in threads other people have asked honest questions in and which you have no clue about the answer. We all know you're an ignorant pathological liar, just keep the evidence in your own threads.
I believe I've found the limit to the first problem, it's \(e^{-1/2}\). I'll double check and post the calculation after, but I gotta run out and meet some friends right now :m:
The latter one is trivial. One application of L'Hopital's Rule does it. A freshman should be able to do it. A freshman might be able to do the former problem. Cpt. Bork: If the limit exists, the limit of the sequence formed by taking the natural log of each element is the natural log of the limit.
None of what you said had any relevance whatsoever to the math question that was asked. It's like if I asked you what the capital of Germany is, and you answered "strawberry ice cream is the best flavour out there!" Indeed, that's the way I handled it. Wasn't too tricky, 1 little subtlety in order to apply L'Hopital's rule, but otherwise a bunch of handstand calculations. This problem makes me want to dig out my old analysis notes. I had an ingenius professor for my first analysis course who knew all kinds of simple, clever tricks for solving these types of problems, whereas others would have to rely on all kinds of fancy calculus and other sophisticated techniques. So did I get the right answer?
It may be known from power series that \(ln (1 + x)\)~\(x\) and \(sin x\)~\(x\) as \(x \to 0\) Therefore: \( {\lim }\limits_{x \to 0} \frac{{\ln \cos x}}{{\sin x^2 }} = \frac{1}{2}{\lim }\limits_{x \to 0} \frac{{\ln (1 - (\sin x)^2 )}}{{\sin x^2 }} = \frac{1}{2} {\lim }\limits_{x \to 0} \frac{{ - (\sin x)^2 }}{{\sin x^2 }} = - \frac{1}{2}\) What does this say about \( {\lim }\limits_{x \to 0} (\cos x)^{\frac{1}{{\sin x^2 }}} \) and the answer to the first question? :shrug:
Can you recheck the first 3 limits? I get -1/2, -1/2, and -1 respectively, whether I use the Taylor expansion approximation or L'Hopital's rule and other methods. Anyhow, if my result is correct, then the 4th limit would be \(\lim_{x\mapsto 0}\left(cosx\right)^{\frac{1}{sin(x^2)}}=e^{-1/2}\). Would you agree?
That's a sweet solution. I need to read up more on ways of applying matrices to do these tricks, I took a Putnam prize seminar and there were some amazing and totally unexpected things you can do with them. In more basic terms, I think one can simply argue that for any chosen number from 1 to n, there are \(2^m\) ways to distribute this number into \(m\) sets/boxes. Since each of the numbers from 1 to n must have a different distribution than all the others, which is a direct consequence of the initial problem's wording, this means that \(n\leq 2^m\). I would say this is equivalent to Temur's reasoning. You count up the number of different ways a given number from 1 to n can be distributed into the m subsets, which can be counted by various methods and gives you a total of \(2^m\). Then you argue that in the worst case scenario, each of these different ways of distributing numbers is used up, and the conclusion follows.
There's a factor of 1/2 in the last limit I wrote. Otherwise you're correct! I'm also interested in seeing how these things can be solved with elementary methods. Let's see your proof of the other one.
Take logs, Taylor expand. I.e: \(\log a = \lim_{x\rightarrow \infty} x\log \left( \frac{ 1}{e}\left(1+\frac{1}{x}\right)^x\right) = \lim_{x\rightarrow \infty}\left(-x + x^2\left( \frac{1}{x} - \frac{1}{2x^2} \right) + o(1) \right)\) So the limit is \(e^{-1/2}\). Ye of little faith...
Perhaps: beauty is in the eye of the beholder I guess! Please Register or Log in to view the hidden image!
Are you actually being serious, or are you having a laugh? I've now read a few of your posts on here and I can't decide which is the more likely!
Here's another one: Let X be a set of positive integers that sum to 100. How large can their product be? I was given this in my first year at uni (is this equivalent to freshman year?).
Ok, I thought you meant you could just eyeball the answer without taking logs and doing Taylor expansions or L'Hopital's rule, etc. I do have to say though, I'm not a big fan of the power series expansion methods because then you have to justify the negation of the higher order terms. But to each their own. Our answers agree Please Register or Log in to view the hidden image!
Usually, first year university is considered freshman. Depends what country, province, state, etc. though, every place has its own little quirks to make it unique. I am going to venture an initial guess of \(2^{50}\) as the answer, but I'll see if I can actually justify it in a bit.
Actually scratch that, I did some thinking and now I believe it's \(4\times 3^{32}\), but I'll work on a proof before I say more.
I've tried to teach you many many times, you don't want to be taught. My point, which whistled over your head, was that your answer was nonsense. Someone asked a maths question and you answer back with physics. Where, in the original post, did anyone mention physical units? And yet you mentioned mass. What did tachyons have to do with it? How is '+' a period? How can you possibly think people believe anything which vomits out of your delusional mouth? Honestly, do you have emotional problems? I cannot think of any other reason you'd lie so repeatedly when you know you'll be caught every time.
Ok, so I think I have it now. The question is: given a set of numbers which sum to 100, what is the largest possible product they can form? My answer: the largest possible product is \(4\times 3^{32}\). My (attempted) proof: Suppose I have a set of numbers \(A={a_1,a_2,\ldots,a_k}\) which sum to 100 and which I claim gives the largest possible product. We know that for \(n>4\), the inequalities \(2(n-2)>n\) and \(3(n-3)>n\) are both true. So if there exists an element \(a_i\in A\) such that \(a_i>4\), I may then write \(a_i=2+(a_i-2)\) or \(a_i=3+(a_i-3)\), and I would thereby produce a set of numbers summing to 100 but with an even larger product, contradicting my claim to have found the maximal product. By contradiction, we must have \(a_i\leq 4\ \forall \ i=1,2,\ldots,k\). Now to simplify things a bit, if there exists an element \(a_i\in A\) such that \(a_i=4\), I may set \(a_i=2+2\) without changing the net sum or the net product. So we deduce that the summation whose components have the largest product is of the form \(100=2+2+2+\ldots+2+3+3+3+\ldots+3\). Now if I have a triplet of \(2\)'s, we can replace them as follows: \(2+2+2=3+3\) We'd then end up with a larger product, since \(2^3<3^2\). So now we're down to the following two options: \(100=2+3+3+3+\ldots+3\), or \(100=2+2+3+3+3+\ldots+3\). The first option is ruled out, since 3 does not divide 98. Thus the second summation is the one whose components give us the maximal product, and that maximal product is thus \(4\times 3^{32}\). Agreed? Edit: I neglected to consider the case where \(1\) is a component of the sum, but it can easily be shown that it produces a less than maximal product. If we write \(n=1+(n-1)\), then obviously \(1\times(n-1)<n\).
It looks to me like a desperate and ill-fated stab in the dark to hopefully answer a question and give the false appearance that he has a strong math and physics background. It's the only way I can understand why he did what he did.
