How can I prove $\lim_{x\to \infty} f(x) = L$ ? I tried to prove by L'Hospital's Rule, but I only proved when the limit exists... How can I prove when I am not provided that the limit exists?
Let $f:(0,\infty) \to R$ be differentiable. Suppose that $\lim_{x\to\infty}(f(x)+f'(x))=L$. Show that $\lim_{x\to\infty} f(x) = L$ and $\lim_{x\to\infty} f'(x) = 0$. (Hint: Write $f(x) = e^xf(x)/e^x$ and use l’Hopital’s Rule.)    My working for $\lim_{x\to\infty}f'(x)=0$:  For $\lim_{x\to\infty}f'(x) = 0$, I let $f(x) = e^xf(x)/e^x$ and applied quotient rule which then cancels off $e^{2x}$ and I'm left with $\lim_{x\to\infty}(f(x)+f'(x)-f(x))$. Can I then equate this with $\lim_{x\to\infty} \left(f(x)+f'(x)\right) - \lim_{x\to\infty}f(x)$ which then gives $L-L=0$? Is this step correct?
How can I get barcode-scan information into my photo metadata, onsite, untethered? I need a DSLR camera with a barcode scanner for on-site tours photography, which means it needs to be untethered! The image would have to contain the barcode scanned information in the image metadata or in the name itself.  I've seen solutions like, Foolography and a $2600 Canon Studio Edition launched in 2010. But I'm not sure of either those products!  So I'm asking the gurus here how I can get this working, which camera would I need and any accessory necessary?  I'm trying to recreate something like I saw the on-site volume photographers do in Disney's parks. How can the petrification from the Flesh to Stone spell be removed? How to calculate Area Under the Curve (AUC), or the c-statistic, by hand I am interested in calculating area under the curve (AUC), or the c-statistic, by hand for a binary logistic regression model.  For example, in the validation dataset, I have the true value for the dependent variable, retention (1 = retained; 0 = not retained), as well as a predicted retention status for each observation generated by my regression analysis using a model that was built using the training set (this will range from 0 to 1).  My initial thoughts were to identify the "correct" number of model classifications and simply divide the number of "correct" observations by the number of total observations to calculate the c-statistic. By "correct", if the true retention status of an observation = 1 and the predicted retention status is > 0.5 then that is a "correct" classification. Additionally, if the true retention status of an observation = 0 and the predicted retention status is &lt; 0.5 then that is also a "correct" classification. I assume a "tie" would occur when the predicted value = 0.5, but that phenomenon does not occur in my validation dataset. On the other hand, "incorrect" classifications would be if the true retention status of an observation = 1 and the predicted retention status is &lt; 0.5 or if the true retention status for an outcome = 0 and the predicted retention status is > 0.5. I am aware of TP, FP, FN, TN, but not aware of how to calculate the c-statistic given this information. Time reversal symmetry and T^2 = -1 Script for display wrapper class Account list of records? Hi i am trying to write script for display list of records from wrapper class.   @remoteaction     public static wrappercls getaccounts(){         string account = 'select id,name,accountnumber from account';         listacc = database.query(account);         count = listacc.size();         system.debug('#####'+count);         wrappercls wra = new wrappercls(count,listacc);         return wra;     }      public class wrappercls{         public integer count11;         public list&lt;account&gt; account;          public wrappercls(integer count11,list&lt;account&gt; account){             this.count11 = count11;             this.account = account;         }     }    This class returns list of account records and record count parameter.Need to display these records in page.Is it possible if possible Please help me on this..  Script  &lt;script&gt;   function getallaccounts(){     exremote.getaccounts(         function(result,event){             if(event.status){              }             else {              }        },{escape:true}     ); } &lt;/script&gt;   Thanks in Advance... Combining the second elements of a set of ordered pairs based on their first element In C#/VB.NET/.NET, which loop runs faster, for or foreach?  Ever since I read that a for loop works faster than a foreach loop a  I assumed it stood true for all collections, generic collections, all arrays, etc.  I scoured Google and found a few articles, but most of them are inconclusive (read comments on the articles) and open ended.  What would be ideal is to have each scenario listed and the best solution for the same.  For example (just an example of how it should be):   for iterating an array of 1000+ strings - for is better than foreach for iterating over IList (non generic) strings - foreach is better than for   A few references found on the web for the same:               [Edit]  Apart from the readability aspect of it, I am really interested in facts and figures. There are applications where the last mile of performance optimization squeezed do matter. Number of bitstrings with $000$ as substring I have $F_n$ number of bitstrings that have $000$, How would I prove that for $n \ge 4$ , $a_n = a_{n-1} +a_{n-2}+a_{n-3}+ 2^{n-3}$?  Now there are many  ways to go about this but if I choose starting a $n-3$ bit string and append it with $000$ how would I show that? I have a problem with the interpretation of a test result in which the p-value is 8.6e-28. How should it be interpreted? What is the e for?  Specifically, I used White's Test for the homoscedasticity assumption of a linear regression:  White's general test statistic :  150.0535  Chi-sq( 9)  P-value =  8.6e-28 Does GCM (or GHASH) only provide 64-bit security against forgeries? In a  a doubt was voiced about my answer, which claims GCM to requires $2^{128}$ for a successful forgery. The doubt was that the square root needs to be taken meaning the security would be $2^{64}$.  So of course I immediately checked the relevant security theorem ():  $$\mathbf{Adv}^{\text{auth}}_{\operatorname{GCM}[\operatorname{Perm}(n),\tau]}(\mathcal A)\leq \frac{0.5(\sigma+q+q'+1)^2}{2^n}+\frac{q'(\ell_A+1)}{2^\tau}$$  with $\sigma$ being the total plaintext size in blocks, $q$ being the total number of encryption queries, $q'$ being the total number of decryption queries, $n$ being the size in bits of the underlying permutation, $\ell_A$ being maximum authenticated input length in blocks and $\tau$ being the tag size in bits.  Now clearly from this we can see that performing $2^{n/2}=2^{64}$ queries yields a strong enough advantage for a break.  Now my question is: When talking about "n-bits of security" or "needs $2^n$ operations to break", do we usually talk about "online" security, ie queries to an oracle can be done and cost $1$ or "offline" security where no oracles are available or do we need to make this decision depending on the situation?  TL;DR: Does GCM offer 64 or 128 bit security? How to Set Firefox 4 to only store History for a set period of time, i.e. one day? $2 \times 2 $ Orthogonal Group I am trying to prove that $O(2)$ has two connected components. This is what I have done: Suppose $A \in O(2)$. Then $A^tA=I$, where $A^t$ is the transpose of $A$ and $I$ is the identity. Taking the determinant of both sides of this equation, we get $\det(A^t)\det(A) = (det(A))^2$   which implies $\det(A) = \pm 1$.  And since $\det$ is continuous (which means that the inverse image of an open set is open), we have $O(2)$ is the disjoint union of $O^+(2)$ and $O^-(2)$. Is this right? Would this be sufficient or would this only prove that there are at least two components? Prove that: $\sqrt{2\sqrt{3\sqrt{4\cdots\sqrt{n}}}}<3,\,\forall n\in\mathbb N.$ I recently changed my username to Undo. I was amazed when I pressed Submit and saw that it actually went through - surely someone somewhere was using this same username.  Out of curiosity, I then search the SO user list for Undo - and found  with Undo as thier name. They have 50 rep. I have around 1,500.   Is there some kind of pecking order, I.e. a high-rep user can 'steal' a low-rep user's name? Or is this a bug?  Or can two users have the same username without causing the SO servers to go down in a burning ring of fire? How do I install the VirtualBox version from Oracle to install an Extension Pack? Integral $\int_{0}^{\infty}e^{-ax}\cos (bx)\operatorname d\!x$ I want to evaluate the following integral via complex analysis $$\int\limits_{x=0}^{x=\infty}e^{-ax}\cos (bx)\operatorname d\!x \ \ ,\ \ a &gt;0$$  Which function/ contour should I consider ? Storing the original document class page layout with geometry Is it possible, using the geometry package, to store (and restore) the original layout specified by the document class (in my case, the class is report)?    Here is some context.  I have been editing a long document in the default page layout for the report document class.  Now I found out that I need a title page.  There is a particular title page layout, and in fact, it works perfectly if I just load the geometry package with its default and change just two settings.  But then I want the rest of the document back in the original page layout!  I can't do something like \restoregeometry, since I think geometry just overrides the class options... i.e. there doesn't seem to be anything to restore.  Similarly, it seems I can't use \savegeometry or \loadgeometry.  So how can I get back to my original layout for the rest of the document without painstakingly hard-coding in all of the specific page layout variables?    Here is a minimum working example:   \documentclass[a4paper,10pt]{report}  \RequirePackage[a4paper,dvips]{geometry} % g1 \geometry{top=2.5cm,bottom=2.3cm}        % g2  \begin{document}  \input{the_title_page}  \SomeGeometryCommandsHere  % g3  \input{the_rest}  \end{document}   If I run the above code with the lines labeled "g" commented out, then the title page does not have the correct page layout, but the rest of the document is formatted perfectly.    If I leave lines g1 and g2 uncommented, then the title page has exactly the correct layout, but no matter what I put for g3 (e.g. \loadgeometry, \restoregeometry, etc.) I can't recover the original layout that I had when g1 and g2 were commented.  Of course, I could completely specify the original layout exhaustively in line g3, and this should in principle work.  However, I spent almost one hour on this last night and when I still couldn't get it formated correctly I decided I really needed a smarter solution. Contains and Custom Label in Rule Criteria - Workflow I have a picklist field-XYZ which contains a lot of values.Lets say (A,b,c,d...z) I am trying to write a rule criteria which will execute if the picklist value is any of the following (A,b,c...p).So instead of mentioning all the values in rule criteria , i created a custom label and put the values (a,b,c,d..p) in that and wrote the following in the rule criteria :   CONTAINS(TEXT(XYZ__c) ,$Label.CustomLabel))   This is not working even is the value of the picklist is equal to the one mentioned in the Custom Label.I guess the CONTAINS function is matching the whole value of custom label with Picklist.Is there other workaround/solution for this? Sum with binomial coefficients: $\sum_{k=1}^m  \frac{1}{k}{m \choose k} $ I am trying to use the  package.  I have the following counters defined (hopefully this is the right way to do it!)  \newtheorem{thm}{Theorem}[section] \newtheorem{lemma}[thm]{Lemma} \newtheorem{cor}[thm]{Corollary}   This works as I would like (i.e. Theorem 1.1, Corollary 1.2, Lemma 1.3, etc.)  My problem is \cref always returns Theorem 1.2, Theorem 1.3. I   Perhaps this is not right? (In my preamble)  \crefname{thm}{theorem}{theorem's} \crefname{lemma}{lemma}{lemma's}