[Archive!] Pure mathematics, physics, chemistry, etc.: brain-training problems not related to trade in any way - page 113
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Richie, I knew you were good at puzzles. These are all the solutions? Three equations and six unknowns.
I'm not a mathematician, I don't bother. I solve such equations simply - with the computer, the method "switch on and go, and see if you get it" :)
Besides, who says there are three equations? It's one :)
P.S. On the Mechmatov forum the argument about the limit lim ( ln ( 2 + sqrt(arctg ( x) ⋅ sin ( 1/ x ) )), x → 0 ) is not over yet; as arguments they started mentioning Hausdorff topological spaces, which I know nothing about. But people, apart from the remaining two (me along with one other person), think we should acknowledge that there is a limit after all.
I have a big request to Farnsworth and lea. Please check, if you don't mind, such a limit on the same packages as before (Mathematica, Maple, MathCad - on all three):
The sine argument is 1/x, and the limit itself is taken to the right of zero.
Checked in Maple13.
To the left and to the right - does not exist. If the direction is not set - it takes, the answer - ln(2).
Although I would say that for such a limit will be ln(2), because lim(arctan) still equal to zero, while sin(1/x)-1 is bounded.
And in what case is there a limit without direction? When the limits on the left and right are equal?
p.s. And where did the "-1" get added from? Or is this some cunning move that might help the solution?)
p.p.s. I'm starting a semester, I'm going to go ask the teachers questions next week)
Проверил в Maple13.
Слева и справа - не существует. Если направление не задавать - берёт, ответ - ln(2).
Хотя я бы и для такого предела сказал, что будет ln(2), т.к. lim(arctan) всё равно равен нулю, а sin(1/x)-1 ограничен.
А в каком случае существует предел без направления? Когда пределы слева и справа равны?
p.s. И откуда "-1" добавилось? Или это какой-то хитрый ход, который может помочь решению?)
p.p.s. У меня начинается семестр, на следующей неделе пойду задавать вопросы преподавателям)
Thank you, very interesting. And it's very strange that without setting a direction it picks up, even though it doesn't pick up on the left and right. It shouldn't be like that.
I added -1 myself to demonstrate a function that has a limit point in the right neighborhood of zero in the area of definition (zero), but its area of definition itself is countable. That is, the function is not defined almost everywhere (the term "almost everywhere" is quite mathematical and means "everywhere, except no more than a countable set" - of course, if we are talking about an initial set of power continuum).
Have a look here, that's where the whole argument is.
And try to give the first limit to the teachers first, listen, and, if they think it exists, give the second one, with minus one. Draw their attention to the area of definition of the second function.
Загляни сюда, тут весь спор.
I'm already reading it.
Try to give the teachers the first limit first, listen, and, if they think it exists, give the second one, with minus one. Draw their attention to the area of definition of the second function.
OK.
You ask them my questions. Especially about the fly bubble. When I was at university, one associate professor got so jammed up that he still can't forgive me :)
Nah, I still have to study))))
lea does not look like a student who only cheats. Especially if he doubts his ability to take limits and goes back to Fichten. For most students, this is simply a stage that has been passed and there is no need to go through it again, because it is 'fucked'.
lea does not look like a student who only cheats. Especially if he doubts his ability to take limits and goes back to Fichten. For most students, it's just a stage that has been passed and there's no need to go through it again, because it's "fucked".
No, I'm not talking about lea, I'm talking in general. I remember only 3 students in our group passed maths without cheating.
The Philosophy - nobody took it because the senior lecturer M couldn't get to the lecture- he was so drunk that he couldn't even get to the door of the university :)
P.S. On the Mechmatov forum the argument about the limit lim ( ln ( 2 + sqrt(arctg ( x) ⋅ sin ( 1/ x ) )), x → 0 ) is not over yet; as arguments the Hausdorff topological spaces were mentioned, in which I know nothing. But people, apart from the remaining two (me along with one other person), think it should be acknowledged that a limit does exist.
The sine argument is 1/x, and the limit itself is taken to the right of zero.
I think the notion of a limit has to be approached within a definition. And that definition requires actually continuity in the vicinity of the limit point, one- or two-sided. If the root is just arctg*sin, then the limit is undefined, because the sign of the expression is undefined. Although the value at the limit point x=0 is present. If (-1) is involved there, then the limit does not exist because the sub-rooted expression is negative everywhere except at x=0.
IMHO, this is the interesting case where the value is well defined but the limit is not.
Next: Prove that the degree of two cannot end in four identical digits.
Yurixx >> Если там участвует (-1), то предела не существует, поскольку подкоренное выражение отрицательно везде, кроме точки х=0.
Not only x=0. They are all points x(n) = 1/((2n+0.5)*Pi). There are a countable set of them, and they have a limit point.
Next: Prove that the power of two cannot end in four identical digits.
What about fractional degrees?
How do you do this? It just says ln(2) (Maple 13).
And one more question. How do I change the default settings for plotting boundaries? When I refresh a sheet, the Graph View changes. :(