"In 2002, in the week ending September 22nd, Dragon Ball Z was the #1 program of the week on all of television with tweens 9-14, boys 9-14 and men 12-24"
@MartinSleziak They are both solved by $|x-y|+|y-z|=|x-z|\iff y$ is between $x$ and $z$. I figured that a difference can be written as a sum, but if enough people think otherwise, it can be reopened. Can you not vote to reopen because I closed it?
Yes I can. But I thought it is useful to hear your opinion.
And also to point at you for unilateral closure...
I am not sure to which extent the users are against unilateral closures by mods. I think I remember some discussion about mods posting their votes in comments, and if there are enough of them, then they use mod powers.
But I might be mistaken (maybe I mixed it with something else).
@MartinSleziak It was noted by someone else, and I agreed. That was a couple of years ago. I think the complaints about unilateral closures started later, or I would not have done it (but I could be mistaken).
In any case if the question you closed is a duplicate, then so are the two questions I have mentioned. (They are much closer, only numbers are different.)
@robjohn Good to know. I did not know what is the position of the mods on unilateral closure now. (But I think I have seen a few quite recently. Although I might have mixed them up with the new golden-badge-dupe-hammer.)
@BalarkaSen I'm deeply sad, I lost the light of my life, a very clever, happy dog, smarter than half of the people I ever met and probably more honest than 99% of the people I ever met. If I should have paid more attention to her, maybe I could have done something, but I did mathematics. I hate myself for that and I don't wanna do mathematics anymore.
5
There is no thing in the world that I could have accepted for giving up that dog,no ammount of money, no book, no achievement, I don't give a s**t on all of them when it's about the life of someone I care about, the only being that made me happy. I'm only mad now.
hi all. i'm revising some material on limits, a subject that i openly find intimidating. i'm trying to understand when to use various strategies. for starters, is it correct to say: 1) if limit is 0/0, try to factorize. 2) if limit is 0/0 with roots involved, use difference of squares. 3) if limit is 0/0 or infinity / infinity (not defined basically), use l'hopital?
If it is monotonically increasing, you can simply rewrite $s_n$ as an integral of piecewise constant function. This function lies bellow $f(x)$ on each interval of the length $1/n$.
@topper. I guess one tries to factorise as best they can. Furthermore, one should try and implement any elementary manipulations to try and make the expression have an obvious limit. If none of these work and the numerator/denominator are continuous, then you can use L'Hopitals.
An interesting example though is $y=\frac{x^2}{x}$.
@GustavoMontano thanks. i guess what i'm really asking, is, if l'hopitals works for all 0/0 situations, and one has a reasonable handle on differentiation, is there any reason not to use l'hopitals straight away, rather than get involved with factorizing? sorry if this is a lame question
okay, i have 2x/1 when i differentiate numerator and denominator. what's the problem? i know the derivative of the denominator can't equal zero, but it's 1...
for example, it's good for me to know that i can use l'hopital for any limit of type 0/0, without having to think about factorizing and other steps. i can just dive in and start differentiating
It's more of a safety net, if anything. It's good to think of different ways to tackle different problems. That's why integrals are pretty neat. There's no algorithm for tackling the problem.
Oh, sorry. Yea, it goes by different names. It's like setting up an integral in terms of a number that can be expressed in terms of a number that's less than the original number. i.e. $I_n = 5I_{n-1}$ where $I_n$ is the original integral, @Nick. The whole point is to eventually reduce your original integral down to one that you can evaluate on it's own. Like how the Gamma function ends up as a bunch of terms multiplied by the integral of $e^{-x}$.
Well, integration by reduction formula usually involve repeated integration by parts. In that sense, you'd be more likely to be asked to find $\int x^{n} \sin x \text{ d}x$.
@Nick $$ \begin{aligned} \int x^n \sin x \text{ d}x \ & \overset{\text{I.B.P}_1}= -x^n \cos x - \int -nx^{n-1} \cos x \text{ d}x \\ & = -x^n \cos x + n \int x^{n-1} \cos x \text{ d}x \\ & \overset{\text{I.B.P}_2}= -x^{n} \cos x + n \left( x^{n-1} \sin x - (n-1) \int x^{n-2} \sin x \text{ d}x \right) \\ & = -x^{n} \cos x + n x^{n-1} \sin x - n(n-1) \int x^{n-2} \sin x \text{ d}x \end{aligned} $$ Now if we call our original integral $I_{n}$, then our final answer will be in terms of $I_{n-2}$.
That's a fairly basic example of integration through deriving a reduction formula. The whole essence of the process is to use integration by parts (or some neat tricks) to reduce an integral down to one that we can evaluate easily, @Nick.
@GustavoMontano Dark Roasts have beans that are roasted to higher temperatures, so they loose most of their caffeine content (which iirc is responsible for the no-sleep thing)
i'm looking for a formal description of when a factor is moved outside the limit to ease calculation e.g. lim x->infinity 1/2 (ln 2-x), the 1/2 can be moved outside the limit to give 1/2 lim x-> infinity ln (2-x). what is this operation called?
preferably a web reference that i can summarize in my notes
