I'm trying to solve this limit:     $$\lim_{x\to-\infty} xe^x$$   I'm trying to solve this limit using the l'Hôpital rule. My question is: can I use this rule in the last limit below?  $\lim_{x\to-\infty} xe^x=\lim_{x\to-\infty} \frac{x}{e^{-x}}$  Note the numerator tends to $-\infty$, while the denominator tends to $\infty$, can I use the l'Hôpital rule in this case?  Thanks
Intuitively, $\lim\limits_{t\to -\infty}te^t=0$, since $e^t\to 0$ much faster than $t\to -\infty$.   Is there a way to more rigorously compute this? Writing it as $e^t/(1/t)$ seems resistant to l'Hopital's rule.
What is the best way to add options to a select from a JavaScript object with jQuery? Input form label alignment top or left? On Takodana,      the main characters and various bar patrons can see the planet Hosnian Prime being destroyed by Starkiller Base with their (respective) naked eyes. How did they see this? Is the Hosnian system close enough to see? If it's not very close and they could just see the explosion, how did the light from it reach them so quickly? Fire (at, on, in, to) target How to correctly choose the preposition in "fire (at, on, in, to) target"? Global and local variables in R How to show $\lim_{n \to \infty} a_n = \frac{ [x] + [2x] + [3x] + \dotsb + [nx] }{n^2} = x/2$? Multiple AES Key Derivation from a master key I saw this question  but it doesn't have a full solution. What is the full solution to the 99 bottles of beer on the wall for Vim adventures?  Context: for the sake of being a good SO member I will give more details but a full solution will answer my question. It seems from reading the other post that my register is incorrect for some reason. I can't get the &lt;NL&gt;. I think that's the issue since pasting won't work... Increasing extent of Raster? Detect init system using the shell Prove that for every odd integer n, there is a group... Prove that for every odd integer n, there is a group with exactly n elements of order 2.  I am new to abstract algebra and proofs in general. Proving existence is difficult for me, but I will show what I have.  $$n = 2k + 1$$ $$k \in \mathbb{Z^+}$$  $$\exists G \supset H = (a_1 , a_2 , ..., a_n, e)$$  $$s.t. |a_1| = |a_2| = ... = |a_n| = 2$$  My instinct is to use Lagrange's Theorem, although I would like to try another method to prove this as we have not gone over cosets in class yet. If someone could just give me a hint of how to start, it would be very beneficial. Thank you. Isomorphism $H_n(D^n, S^{n - 1};A)$ and $A$ The homomorphism $a \mapsto [[a \cdot f]]$ is supposed to be an isomorphism between $A$ and $H_n(D^n, S^{n - 1};A)$ for any homeomorphism $f: \Delta^n \to D^n$. I have proven that it is well defined and injective, but why is it surjective?  The group $H_n(D^n, S^{n - 1};A)$ is generated by elements of the form $a \cdot \sigma + b \cdot \tau + c \cdot \partial(\kappa)$ where $\sigma: \Delta^n \to D^n$, $\tau: \Delta^n \to S^{n - 1}$ and $\kappa: \Delta^{n + 1} \to D^{n + 1}$ and I do not see how an element of this form represents the same equivalence class of $[[m \cdot f]]$ for some $m \in A$. We can write $\sigma$ as $f f^{-1} \sigma$ to get $f$ in there, but I don't see that working out. Any hints? : So how many named special cases of each modal verb are there, in toto? There's   1. subjunctive should, and   2. mandative should, and   3. , at least;   and I s[up]pose 4. Should we? is suggestive should, and   5. Shouldn't we? is tag **should**, and I can think of a lot more.    I ask about only operative uses of this modal verb including literary and formal usage,  but please omit anything obsolete. I'm trying to classify them and learn the formal terms describing each. Are there any apt others in addition to the ones above? (I added the enumeration below)  1. What's 'subjunctive should'?  Whenever should is conjugated in the subjunctive mood? 5. What's a tag **should** ? 6. Where does  should/would  feature? What's it called? This should is detailed in   or  or . The new hot questions algorithm sometimes recurses The hot questions algorithm . I like the effects, since I see lots of new interesting stuff every day rather than waiting 6-8 weeks varying amounts of time for the list to update.  However, the new algorithm also has the interesting side effect of recursing the same question:    Of course, this is a world-shattering workflow-breaking unicorn-hating waffle-despising ERROR tremendously minor matter. But, if the SE devs have opportunity during their amazingly large amounts of free time, it might be nice to add a check for the current page before populating the hot questions list. Prove that if $d$ divides $n$, then $2^d -1$ divides $2^n -1$ Prove that if $d$ divides $n$, then $2^d  -1$ divides $2^n  -1$.  Use the identity $x^k  -1 = (x-1)*(x^{k-1}  +  x^{k-2}    + \cdots + x +1)$ Can you concentrate on a special ability while also concentrating on a spell? There exist several abilities that have the text similar to, this effect lasts while you maintain concentration (as if you were concentrating on a spell). One such example is the Bard College of Glamour: Mantle of Majesty ability.     At 6th level, you gain the ability to cloak yourself in a fey magic   that makes others want to serve you. As a bonus action, you cast   command, without expending a spell slot, and you take on an appearance   of unearthly beauty for 1 minute or until your concentration ends (as   if you were concentrating on a spell). During this time, you can cast   command as a bonus action on each of your turns, without expending a   spell slot.   Another example is the Cleric Trickery Domain: Channel Divinity: Invoke Duplicity ability.     Starting at 2nd level, you can use your Channel Divinity to create an   illusory duplicate of yourself.      As an action, you create a perfect illusion of yourself that lasts for   1 minute, or until you lose your concentration (as if you were   concentrating on a spell).   While you are concentrating on maintaining such an ability would you also be able to maintain concentration on a spell that requires concentration?  There does not seem to be any RAW that limits this however the limitation around concentrating on a second spell exists which may suggest that the intent is that a character cannot concentrate on 2 effects at the same time unless explicitly stated. Prove that $i^i$ is a real number What is potential energy truly? I have a problematic question for which I have been unable to attain a satisfactory answer. What is  truly?   -  I have read about how potential energy can be seen as the "highering" of an object within a field, be it gravitational, electromagnetic, etc.   In a way, an object is lifted to a higher position, and when it eventually falls down again energy is released back.  The object does not even have to be "lifted" beforehand, as with the case of asteroids.   However, since energy cannot be destroyed, it must somehow be stored, yet how does energy truly store itself within a field? It seems hard for me to grasp that energy can physically be stored within the positioning of an object?  I can't really get my head around how a field conserves energy within it. Kinetic energy on the other hand, can be, for me, intuitively understood, as the object really is "moving" and energy manifests itself through motion.  Can anyone elaborate on this, perhaps give reasoning as to why a field can in fact "store" energy? Please point out any fallacious or naive views I seem to hold. Theoretical Proof of Pauli's Exclusion Principle How can I contribute to Ubuntu? I would like to know what kinds of contributions someone can make to Ubuntu. Such as Programmer, User, Proofreader, etc.