Mathematician Chat V.floor(π)

[Number-41]

[QUOTE=Athlias;46103403]

To check this, think of it numerically. Look at the limit to the left and to the right of the x value you're looking for, in this case infinity. I know infinity isn't really a number, so let's say infinity is 99999.5. Put a value lower than that, 999998 into the equation, which would be the left side. What do you get? Undefined. Put 999999 in the equation, the right side. Also undefined. Therefore the limit as x approaches i finity is also undefined.

[/QUOTE]

This one would come out of your test as being undefined at "infinity", while it clearly is not.
[IMG]http://quicklatex.com/cache3/ql_9b11a0c4adb76ca3ac4373f7ef6d2d7c_l3.png[/IMG]

[biccboi]

You can't use l'Hospital (well, you can, but it's not allowed)

[ClosetSkeleton]

Well if you don't mind dirty, ugly math. You can evaluate the first limit knowing that those constants would be "absorbed" by infinity, essentially making the problem [;\frac{x+\sqrt{-x}}{x+\sqrt{-x}};], so you get one.

As for the second one without L'Hopital's rule, hm. Alright, that took me a while, turns out it's not too bad if you divide everything by x. Then the only weird limit is [;\frac{arcsin(x)}{x};] , but if you manhandle that to look like [;\frac{x}{sin(x)};], it's pretty easy.

Third one I really liked, because it requires all those trig identities you like to forget. Take the cosine of each side, remember that cosine is an even function, so you can then get rid of the negative inside the right, use your double angle formula, add, and you're pretty much there.

What class is this for/ where do you go? Just curious.

[Joey90]

[QUOTE=JohanGS;46096284][IMG]http://latex.codecogs.com/gif.latex?%5Clim_%7Bx%5Crightarrow%20%5Cinfty%7D%20%5Cfrac%7B%5Csqrt%7Bx%5E2+1%7D+%5Csqrt%7B1-x%7D%7D%7Bx+%5Csqrt%7B2-x%7D%7D[/IMG]
Find the limit
[/QUOTE]

As is typical with limits we only care about the largest powers of x, so discard the small square roots to get

[;\lim_{x\rightarrow\infty} \frac{\sqrt{x^2 + 1}}{x}};]

Then the leading term on top and bottom is just 'x' (for the top, the 1 gets irrelevant as x gets large) thus the limit is 1.

[QUOTE=JohanGS;46096284]
[IMG]http://latex.codecogs.com/gif.latex?%5Clim_%7Bx%5Crightarrow%200%7D%20%5Cfrac%7B2x%20-%20arcsin%28x%29%7D%7B3x+tan%282x%29%7D[/IMG]
Find the limit
[/QUOTE]

Now we only care about the smallest power of x.

[;sin(x) \sim x;] for small x, thus [;arcsin(x) \sim x;] for small x.

tan has the same approximation, so we can approximate it as

[;\lim_{x\rightarrow\infty}\frac{2x-x}{3x+2x}=\frac{x}{5x}=\frac{1}{5};]
[QUOTE=JohanGS;46096284]
[IMG]http://latex.codecogs.com/gif.latex?arccos%5Cleft%20%28%20%5Cfrac%7B1-x%5E2%7D%7B1+x%5E2%7D%20%5Cright%20%29%20%3D%20-2%20arctan%28x%29[/IMG]
For which real x is this true
[/QUOTE]
To work out where they meet, sketching a graph is a good idea, I won't post here, but it let's you know where to look.

The obvious solutions are x=0,-1 the other asymptotic solution (which is presumably what you're interested in) is at -infinity.

[; arctan(-\infty) = -\pi/2 ;]
[; arccos(-1) = \pi ;]

thus the two sides are equal in the limit.
[QUOTE=JohanGS;46096284]
[IMG]http://latex.codecogs.com/gif.latex?n%5E%7Bn+1%7D%20%3E%20%28n+1%29%5En%20%5C%3B%20%5C%3B%20%5C%3B%20%5C%3B%20%5C%3B%20n%20%5Cin%20%5Cmathbb%7BN%7D%2C%20n%20%5Cgeq%203[/IMG]
Prove by induction[/QUOTE]
Eh, I can do it by doing a binomial expansion, but it's not that neat and I don't want to write out the latex :v:

[Falubii]

Wow I overcomplicated that.

[Number-41]

What plugin do you guys use for formulas? Because they don't show up at all...

I have Tex the World for Chromium (1.3.2)

[Athlias]

[QUOTE=Number-41;46107800]This one would come out of your test as being undefined at "infinity", while it clearly is not.
[IMG]http://quicklatex.com/cache3/ql_9b11a0c4adb76ca3ac4373f7ef6d2d7c_l3.png[/IMG][/QUOTE]

I suppose I didn't make it clear enough. The method I was talking about is for people who might have trouble thinking about things like subtracting from infinity. So as to make it simpler they can think of a relatively big number to put in. It could be 999999 or it could be 9^345, it doesn't really matter as long as it is appropriate for the problem. The concept is that at these high values the function should approach some value or doesn't exist at all. I hope this clarifies what the purpose of the method was supposed to be.

Also checking my work now for the second one I realize I really dropped the ball on that one, sorry about that. I'll make sure to clearly double check my work if I ever happen to post here again.

[Falubii]

[QUOTE=Athlias;46108949]I suppose I didn't make it clear enough. The method I was talking about is for people who might have trouble thinking about things like subtracting from infinity. So as to make it simpler they can think of a relatively big number to put in. It could be 999999 or it could be 9^345, it doesn't really matter as long as it is appropriate for the problem. The concept is that at these high values the function should approach some value or doesn't exist at all. I hope this clarifies what the purpose of the method was supposed to be.

Also checking my work now for the second one I realize I really dropped the ball on that one, sorry about that. I'll make sure to clearly double check my work if I ever happen to post here again.[/QUOTE]

Sometimes it's more useful to see the ways things don't work.

[Joey90]

[QUOTE=Number-41;46108915]What plugin do you guys use for formulas? Because they don't show up at all...

I have Tex the World for Chromium (1.3.2)[/QUOTE]
Exactly that, not sure why it wouldn't work for you.

I've not exactly done it rigorously, although all the steps can be made to be. Depends how detailed it needs to be, or whether you just want to get the answer (by hand).

[biccboi]

Oh, the first limit goes to negative infinity but it's not really that hard. The third requires some evaluation of domains and eventually you get that it's true for all negative real x.

[QUOTE=Xeloras;46108541]Well if you don't mind dirty, ugly math. You can evaluate the first limit knowing that those constants would be "absorbed" by infinity, essentially making the problem [;\frac{x+\sqrt{-x}}{x+\sqrt{-x}};], so you get one.

As for the second one without L'Hopital's rule, hm. Alright, that took me a while, turns out it's not too bad if you divide everything by x. Then the only weird limit is [;\frac{arcsin(x)}{x};] , but if you manhandle that to look like [;\frac{x}{sin(x)};], it's pretty easy.

Third one I really liked, because it requires all those trig identities you like to forget. Take the cosine of each side, remember that cosine is an even function, so you can then get rid of the negative inside the right, use your double angle formula, add, and you're pretty much there.

What class is this for/ where do you go? Just curious.[/QUOTE]

Those are some tasks from a practice exam for the introductory analysis course at Chalmers University of Technology (specifically for the Engineering Physics programme, and also Mechanical Engineering this year).

[Krinkels]

Is there a way I can prove that pi is an infinite sum of positive rational numbers without actually constructing such a series?

[swebonny]

Guys make a new thread :v:

[editline]3rd October 2014[/editline]

Here's the source of the OP
[quote][noparse]
I waited for a while for some more knowledgeable user to create a thread, but nope. So I'm making it instead.

Some resources I've been using:

[B]WolframAlpha[/B]
[URL]http://www.wolframalpha.com[/URL]

[B]List of Math Symbols
[/B][URL]http://en.wikipedia.org/wiki/List_of_mathematical_symbols[/URL]

[B]Paul's Online Math Notes, really useful Calculus I-III, Linear Algebra and Differential equation notes
[/B][URL]http://tutorial.math.lamar.edu/[/URL]
[B]
Khan Academy, math videos with easy(non-formal) explanations
[/B][URL]https://www.khanacademy.org/[/URL]

[B]Latex Cheat Sheet:
[/B][URL]http://www.stdout.org/~winston/latex/latexsheet-a4.pdf[/URL]

[QUOTE=Bradyns;39766368]Swebonny, here are some to add..

[B]Youtube users:[/B]
[URL="http://www.youtube.com/user/patrickJMT"]PatrickJMT[/URL]
[URL="http://www.youtube.com/user/TheIntegralCALC"]TheIntegralCALC[/URL]

[B]Websites:[/B]
[URL="http://www.mathtutor.ac.uk/"]Maths Tutor [UK][/URL]
Awesome videos from simple algebra to various integration techniques.

[URL="http://sites.google.com/site/scienceandmathguide/subjects/mathematics"]Mathematics - /sci/[/URL]
/sci/ board made their own page, which is a wealth of knowledge.[/QUOTE]

[QUOTE=Roll_Program;39765005][URL]http://www.mathpages.com/[/URL]

Some extremely useful stuff, especially for physics.[/QUOTE]

[QUOTE=JohanGS;39773807]I like to use this when I need post problems here, you can use the created picture without having to upload it or anything

[URL]http://www.codecogs.com/latex/eqneditor.php[/URL][/QUOTE]

I'll add more if you give me some. Most of the times [B]Wikipedia [/B]is all you need.
I'm currently taking a Probability theory and Statistics course, pretty damn interesting and fun.
[/noparse][/quote]

[JohnnyMo]

[QUOTE=Swebonny;46139808]Guys make a new thread :v:

[editline]3rd October 2014[/editline]

Here's the source of the OP[/QUOTE]

As you wish.