All along, I had the impression that, as reward for resisting the One Ring and contributing towards the journey to its ultimate demise, Bilbo (for finally abandoning it), Frodo (for having done the most to carry the burden for the actual journey) and Sam (for carrying it when Frodo couldn't and successfully resisting temptation to take over as the main bearer) were granted passage to spend the remainder of their mortal days in the Undying Lands, where the Blessed Realm can help heal/protect their souls from the dark taint of the Ring until the end of their days.  I just found Arwen's wiki article, though, and a passage quotes thus:     After marrying Aragorn, she bestowed upon Frodo Baggins a gift: Her place on the ships to Valinor.   This implies that Frodo was originally ineligible for entry into the Undying Lands, until Arwen gave him her ticket.  Is my prior impression wrong? I think it came from many years ago, the one time I read RotK. If I am correct, then why does the quoted passage seem to conflict it? Did I misinterpret it, or is it a movie vs book thing? If I am wrong and the quoted passage is correct, then:   Why can Bilbo and Sam go to the Undying Lands? Whose place on the ships did they take? Arwen already gave up her immortality to be with Aragorn. She is known to have told her father Elrond: "There is now no ship that would bear me hence". I don't think the choice of abandoning immortality can be walked back once made, so why did she have a seat to offer to anyone by that point? Even if she still held a place on the ships to the West, is it her place to offer it to anyone?
At the end of The Return of the King, Frodo, Bilbo, Gandalf, Elrond, and Galadriel left Middle-earth on a ship, along with other elves. I named those five because they were all ringbearers and it seems to be a common trait. Frodo and Bilbo bore the One Ring and the three others bore the three elven Rings.  Why did they have to do that? Why didn't they stay and have a peaceful life now that the Dark Lord was gone?
Give each site a parallel site for polling, recommendations and subjective-ish stuff Stuff like polls, recommendations based on subjective constraints, puzzles, webcomics etc. do not belong on the serious main SE sites, where professionals should be considered at work and having just a few spare minutes ("code's compiling") to answer questions, so they should not be distracted by such things. However, I'd also like to have a home for these things still using the SE engine. For the reason I mentioned before, this needs to be a separate place though. Let's call it four.[sitename-here].com. Problem:  If $\gcd(a, b) = 1$ and If $ab = x^2$ ,prove that $a$, $b$ must also be perfect squares; where $a$,$b$,$x$ are in the set of natural numbers  I've come to the conclusion that $a \ne b$ and $a \ne x$ and $b \ne x$ but I guess that won't really help me.. I understand that if the $\gcd$ between two numbers if $1$ then they obviously have no common divisors but where do I go from this point?  Any tips at tackling this would be great. It looks quite easy though I'm still trying to get my hand around these proofs! Any pointers in the right direction would be great.  Thank you in advance, I am reviewing take aways. I am having trouble  How do I do $342 - 58$?  For the ones column I made the $2$ into a $12$ so I can do $12 - 8 = 4$ but I must take away one tens. So I get $3 - 5$ in tens column but I cant do $3 - 5$. What do I do now do I borrow something else? What are the 'types' of parametric down conversion? In previous versions of the OS in the security pane, you are provided an option of:    But in this version there is no option anywhere:    Is there a way that I can turn on anywhere? (via Terminal maybe?) How does the Doctor know how old he is? I've often wondered, how does the Doctor know how old he is? He not only travels through time, but also through space, and visits planets where days can be shorter or longer than 24 hours. How does he 'keep score'?  Also, does he count his age in Earth years or Gallifrey years? They might be completely different in length. What is the life expectancy of a Wookiee? At lunch, my coworkers and I were discussing The Force Awakens, and started questioning Chewbacca's age.  A quick check of  and it says he is possibly about 230 years old as of Episode VII.  Following the  link on that page, Wookieepedia claims that they live to be 400 years old.  Do we know from canon what the average life expectancy of a Wookiee is? Difference between [NSMutableArray array] vs [[NSMutableArray alloc] init] How can I take browser screenshots at a higher resolution than my browser supports? In JavaScript can I make a "click" event fire programmatically for a file input element? Prove that in any GCD domain every irreducible element is prime What happens to a polymorphed character if he is shorn as a sheep? what is the general definition for some partial differential equation being called elliptic, parabolic or hyperbolic - in particular, if the PDE is nonlinear and above second-order.  So far, I have not found any precise definition in literature. Is there any danger to letting food cook in a slow cooker for a very long time? I've recently started using a slow cooker and I was thinking about recipes that could be cooked for days.  My main concern is if there are any side effects for leaving food cooking for days that would make this plan unfeasible (dangerous to try).  Since it is at cooking temperature, there wouldn't be any reason to worry about bacteria growing on the food.  Liquid would be lost over time, which could result in burning if left unchecked, but if I add more every morning/night as needed, that too wouldn't be a problem.  All I can see is that most recipes wouldn't end up tasting as well, but undesirable isn't equal to dangerous.  There just seems to be something absurd with the notion of leaving food cooking for 5 days or even 2 weeks without there being any dangers, but I'm having a hard time figuring out what they would be. What is the unisex form of a word like fisherman? Do you have to use fisherman and fisherwoman separately, or is fisherperson acceptable? I couldn’t find a dictionary with the word …  In general, what do you do when a word does not have a unisex form? How do I destroy Nekker nests? I was about to add an extra signal handler to an app we have here and I noticed that the author had used sigaction() to set up the other signal handlers. I was going to use signal(). To follow convention I should use sigaction() but if I was writing from scratch, which should I choose? $X/{\sim}$ is Hausdorff if and only if $\sim$ is closed in $X \times X$ How to implement custom icons for my script/addon? I am able to evaluate the limit $$\lim_{x \to 1} \frac{x + x^2 + \dots + x^n - n}{x - 1} = \frac{n(n + 1)}{2}$$ for a given $n$ using l'Hôspital's (Bernoulli's) rule.  The problem is I don't quite like the solution, as it depends on such a heavy weaponry. A limit this simple, should easily be evaluable using some clever idea. Here is a list of what I tried:   Substitute $y = x - 1$. This leads nowhere, I think. Find the Taylor polynomial. Makes no sense, it is a polynomial. Divide by major term. Dividing by $x$ got me nowhere. Find the value $f(x)$ at $x = 1$ directly. I cannot as the function is not defined at $x = 1$. Simplify the expression. I do not see how I could. Using l'Hôspital's (Bernoulli's) rule. Works, but I do not quite like it.   If somebody sees a simple way, please do let me know.    Added later: The approach proposed by Sami Ben Romdhane is universal as asmeurer pointed out. Examples of another limits that can be easily solved this way:   $\lim_{x \to 0} \frac{\sqrt[m]{1 + ax} - \sqrt[n]{1 + bx}}{x}$ where $m, n \in \mathbb{N}$ and $a, b \in \mathbb{R}$ are given, or $\lim_{x \to 0} \frac{\arctan(1 + x) - \arctan(1 - x)}{x}$.   It sems that all limits in the form $\lim_{x \to a} \frac{f(x)}{x - a}$ where $a \in \mathbb{R}$, $f(a) = 0$ and for which $\exists f'(a)$, can be evaluated this way, which is as fast as finding $f'$ and calculating $f'(a)$.  This adds a very useful tool into my calculus toolbox: Some limits can be evaluated easily using derivatives if one looks for $f(a) = 0$, without the l'Hôspital's rule. I have not seen this in widespread use; I propose we call this Sami's rule :).