Could somebody give me the answer: Is it possible to prove that $\lim_{x\to 0} {\ln(x)x} = 0$ without L'Hospital's rule?
I have a midterm coming up and on the past exams the hard question(s) usually involve some form of $\lim_{x\to0^{+}} x \ln x$. However, we're not allowed to use l'Hopital's rule, on this year's exam anyways.  So how can I evaluate said limit without l'Hopital's rule? I got somewhere with another approach, don't know if it's useful:   $\lim_{x\to0^{+}} x \ln x = \lim_{x\to0^{+}} x^2 \ln (x^2) = L$ $= (\lim_{x\to0^{+}} 2x)(\lim_{x\to0^{+}} x \ln x)$ $= 0 * L$   Then I just need to prove that L is finite/exists (which means it must be 0)
Coefficient changes sign when adding a variable in logistic regression In my logistic regression the sign of coefficients of a variable (location distance of an amenity) changes based on other variables (with time -ve, with travel distance +ve) in the model. When the location distance is the only variable in the model, it has +ve sign.    Should the variable need to maintain the +ve sign no matter what other variables are added in the model? Does changing sign signify a multicollinearity issue? Some IVs are gaining significance while in a bivariate model, they didn't show significance and vice versa. Is it okay to add variables that don't have much significance (ex: travel distance has a significance of 0.33 individually, but 0.05 when added with other variables) but becomes significant in the model? thanks. In the first series of the TV show several things apparently bypass  to reach Winterfell:   The Night's Watch member  captured near Winterfell and executed by Ned. The Direwolf &amp; her pups found immediately after the execution  and the other Wildlings travelling with her   If The Wall is approximately the same along its length it would presumably be impassible. Is the only option for them to go around the western end (since it stops at the Shadow Tower)? Is this clarified in the book? Are other methods mentioned in subsequent books? When i tried to install PIP for python3 in Ubuntu 18.04.2 LTS  Command I used:   root@DESKTOP-0BEE3OD:/mnt/c# apt install python3-pip  E: Could not read response to hello message from hook [ ! -f /usr/bin/snap ] || /usr/bin/snap advise-snap --from-apt 2&gt;/dev/null || true: Success   I am not able to fix this error. How to keep rice in a biryani from burning and sticking together? The bottom part of my  Biryani always  gets burnt, rice always gets sticky but I want each grain separated. How can I prevent burning &amp; stickiness of rice? $x^{p-1} + ... + x^2 + x + 1$ is irreducible using Eisenstein's criterion? Eisenstein's criterion says that for a prime $p$ if the following conditions are satisfied for a primitive polynomial, $f(x)$, then that polynomial is irreducible in $\mathbb{Z}[x]$  $p \mid a_0, a_0, ..., a_{n-1}$   $p \not\mid a_n$  $p^2 \not\mid a_0^2$  Now I am almost certain I read somewhere that this lemma can be used to prove that the polynomial $x^{p-1} + x^{p-2} + ... + x^2 + x + 1$ is irreducible.  All coefficients are $1$ though and we have have an awkward amount of terms, $p - 1$ terms as opposed to $p$ terms - so how can Eisenstein's criterion be applied here? '\r' added end of the script command This is my script:  #!/bin/bash hd='pwd' cd $hd   When I try to run it, it fails with:  line 3: cd: $'pwd\r\r': No such file or directory   What is going on? Are there functions or packages in Mathematica to sort subsets in lex order, colex, revlex, etc.? Custom environment in wrapfigure I have some conditions for some background job to run:  condition-command &amp;&amp; condition-command &amp;&amp; background-job &amp;   The problem is: I want the conditions to block until the job runs, as if I had run:  condition-command; condition-command; background-job &amp;   But it isn't a condition, if previous command fails I do not want the job to run.  I realised it is asynchronous, but it should not, in my mind the both following scripts should be the same, but they do not:  sleep 2; echo foo &amp; sleep 1; echo bar; wait   # prints foo, then bar: correct sleep 2 &amp;&amp; echo foo &amp; sleep 1; echo bar; wait # prints bar, then foo: bug   I know if I test $? variable it would work, or if I put the last one inside a subshell (but then I would lost job controls, I want to avoid daemons), but I want to know why bash make it this way, where is it documented? Is there any way to prevent this behaviour?  Edit: Chained ifs is disgusting, that is why I will not accept it as a alternative way. Edit 2: I know a subshell is possible, but it will not work for me, let us imagine I want to run a bunch of commands then wait in the end. It will be possible if i check the existence of /proc/$PID directory, but it would be a pain in the neck if there are several jobs. Edit 3: The main question is WHY bash does it, where is it documented? Whether or not it has a solution is a bonus! How to run Ubuntu without GUI in Virtual Box? How can I keep recent files from appearing in Unity? I sometimes browse erotic media files but I would like to avoid having them appear in files &amp; folders / recent. How can I do that?  The old "making .recently-used.xbel a folder" trick from 10.10 and older doesn't work any more.   Plus I'd prefer to turn off tracking only temporarily.  I also prefer not to use a separate user for this, as switching users sometimes isn't fast enough. Give permission to debian-transmission to write into a www-data directory Difference between "does have" and "has" Possible Duplicate:                 What is the difference between does have and has? For example, compare she does have a car and she has a car. Why are differential equations for fields in physics of order two? What is the reason for the observation that across the board fields in physics are generally governed by second order (partial) differential equations?    If someone on the street would flat out ask me that question, then I'd probably mumble something about physicists wanting to be able to use the Lagrangian approach. And to allow a positive rotation and translation invariant energy term, which allows for local propagation, you need something like $-\phi\Delta\phi$.  I assume the answer goes in this direction, but I can't really justify why more complex terms in the Lagrangian are not allowed or why higher orders are a physical problem. Even if these require more initial data, I don't see the a priori problem.  Furthermore you could come up with quantities in the spirit of $F\wedge F$ and $F \wedge *F$ and okay yes... maybe any made up scalar just doesn't describe physics or misses valuable symmetries. On there other hand in the whole renormalization business, they seem to be allowed to use lots and lots of terms in their Lagrangians. And if I understand correctly, supersymmetry theory is basically a method of introducing new Lagrangian densities too.   Do we know the limit for making up these objects? What is the fundamental justification for order two? An  can be identified by its  together with isolating intervals with rational bounds for its real and imaginary parts. The degree of an algebraic number is the  of its minimal polynomial. There are known algorithms that allow to easily compute sum and product of algebraic numbers in this representation, raise them to a rational power, extract real and imaginary parts, compare them, or evaluate them numerically to an arbitrary precision.  Is there an efficient algorithm that given an algebraic number $\alpha$ in this representation can decide if $\alpha$ can be represented as a sum or product of algebraic numbers of smaller degree? Probability question of hats How do I share samba via Open VPN? Is that possible? I have configured open VPN in my home so that it can be accessible from my office easily.  Now I want to share samba via my open VPN. How can I share samba over open VPN between Ubuntu and Windows?  Home public IP: HHH.OOO.MMM.EEE Home desktop IP: 192.168.10.10 Office public IP: OOO:FFF:III:CCC Office Windows desktop IP: 192.168.1.10 I looked "$\divideontimes$" up on Google and now I know that it's Unicode U+22c7 but when would it be used?  I am guessing that $5 \divideontimes 5 = 25$ and $1$...? Proving that closed (and open) balls are convex Why is pruning not needed for random forest trees? Breiman says that the trees are grown with out pruning. Why? I mean to say that there must be a solid reason why the trees in random forest are not pruned. On the other hand it is considered very important to prune a single decision tree to avoid over fitting. Is there some literature available to read for this reason? Of course the trees may not be correlated but there still will be a possibility of over fitting.