How do I solve? I've tried to multiply and divide by the conjugate cannot advance. $$\lim_{x\rightarrow +\infty} \sqrt{(x-a)(x-b)}-x$$
Hi I have a question regarding finding the values of limit for the following question.  Let $a, b \in \mathbb R$. Find the limit $$\lim_{x\to+\infty}\left(\sqrt{(x+a)(x+b)}-x\right)$$
QM and Renormalization (layman) I was reading Michio Kaku's Beyond Einstein. In it, I think, he explains that when physicsts treat a particle as a geometric point they end up with infinity when calculating the strength of the particle's field as you approach the particle.  First, did I get that part right?  Second, how does (in very simple terms) renormalization try to fix this in Quantum Mechanics? If you can include some type of word picture that would be great. Accepted answer will go to the clearest explanation.  Update  I got a little further in the book (lol) and Kaku talks about using symmetry to remove the divergences in the math. I'd appreciate an answer that incorporates this also. Thanks guys! Cards entering the battlefield simultaneously and "Enter the Battlefield" effects Code coverage for wrapper class is 0% I have written below Wrapper class.  global class BI_VA_AutomationWrapper {      global class BI_VA_AutomationRequest {      webservice String userName;     webservice String emailId;     webservice String buinessUnit;     webservice String orgName;     webservice String country;     webservice String environment;     webservice String myShopNumber;     webservice String BidsId;//for testing     webservice String Type; //for testing  }  global class BI_VA_AutomationResponse {      webservice String responseStatus;     webservice String myShopNumber;     //webservice list&lt;Error&gt; errorMessages;     webservice String  errorMessages;//for testing   } global class BI_VA_Error {      webservice String type;     webservice String message;    }}   And test class is,  @isTest public class BI_VA_AutomationWrapperTestClass  { public static testMethod void unitTest(){      BI_VA_AutomationWrapper.BI_VA_AutomationRequest wrap= new BI_VA_AutomationWrapper.BI_VA_AutomationRequest();        wrap.userName='Test@test.com';     wrap.emailId='Test@test11.com';     wrap.buinessUnit='AH';     wrap.orgName='ORD';     wrap.country='US';     wrap.environment='QA';     wrap.myShopNumber='123456789';     wrap.BidsId='123';     wrap.Type='ResetPassword';              BI_VA_AutomationWrapper.BI_VA_AutomationResponse res= new BI_VA_AutomationWrapper.BI_VA_AutomationResponse();     res.responseStatus = 'Success';     res.myShopNumber = '987654321';     res.errorMessages = 'Test error';      BI_VA_AutomationWrapper.BI_VA_Error error = new BI_VA_AutomationWrapper.BI_VA_Error();     error.type = 'Test';     error.message = 'Test';  }}   Code coverage is 0% only. G acts on X transitively, then there exists some element that does not have any fixed points Let $X$ be a transitive $G$-set. ($G$ acts on  $X$ transitively.) If $X$ is finite and has at least two elements, show that there is some element $g$ $\in$ G which does not have any fixed points; that is, such that $g$$($$x$$)$ $\ne$ $x$ for all $x$ $\in$ $X$  I am trying to use contradiction, but it is not very clear to me why this is true. Tabs in output file written by xelatex and pdflatex are different When should I use a struct rather than a class in C#? Proving at least 99 pairwise sums of the reals are non-negative Suppose, one hundred real numbers are given and their sum is 0.Then how can I prove that at least 99 of the pairwise sums of these hundred numbers are non-negative?  I tried this: Let the real numbers be $a_i$,$1\leq i\leq100$. By the given condition, $a_1+a_2+\dots +a_{100}=0$.Suppose for the sake of contradiction, that at most 98 of the pairwise sums of these given reals are non-negative.I just tried to examine the case when exactly 98 of them are non-negative. I labelled the sums $S_1$,$S_2,\dots S_{4950}$,WLOG assume that sums $S_1,S_2,\dots S_{4852}$ are negative, the rest are non-negative. $$S_1+S_2+\dots +S_{4950}=99(\sum_{i=1}^na_i)=0$$ Case I: $S_{4853}+a_{4854}+\dots +a_{4950}=0$.That simply means that $S_1+S_2+\dots +S_{4852}=0$.But, by the given condition, $S_1+S_2+\dots +S_{4852}&lt;0$,a contradiction.After that, I am not able to make any progress.  Can anyone please suggest a more efficient method to solve this problem.I feel this approach will involve enormous case work. Serializing anonymous delegates in C# I am trying to determine what issues could be caused by using the following serialization surrogate to enable serialization of anonymous functions/delegate/lambdas.   // see http://msdn.microsoft.com/msdnmag/issues/02/09/net/#S3 class NonSerializableSurrogate : ISerializationSurrogate {     public void GetObjectData(object obj, SerializationInfo info, StreamingContext context)     {         foreach (FieldInfo f in obj.GetType().GetFields(BindingFlags.Instance | BindingFlags.Public | BindingFlags.NonPublic))             info.AddValue(f.Name, f.GetValue(obj));     }      public object SetObjectData(object obj, SerializationInfo info, StreamingContext context,                                 ISurrogateSelector selector)     {         foreach (FieldInfo f in obj.GetType().GetFields(BindingFlags.Instance | BindingFlags.Public | BindingFlags.NonPublic))             f.SetValue(obj, info.GetValue(f.Name, f.FieldType));         return obj;     } }     Listing 1 adapted from   The main issue I can think of that might be a problem is that the anonymous class is an internal compiler detail and it's structure is not guaranteed to remain constant between revisions to the .NET Framework. I'm fairly certain this is the case based on my research into the similar problem with iterators.  Background  I am investigating the serialization of anonymous functions. I was expecting this not to work, but found it did for some cases. As long as the lambda did *not&amp; force the compiler to generate an anonymous class everything works fine.   A SerializationException is thrown if the compiler requires a generated class to implement the anonymous function. This is because the compiler generated class is not marked as serializable.  Example  namespace Example {     [Serializable]     class Other     {         public int Value;     }      [Serializable]     class Program     {         static void Main(string[] args)         {             MemoryStream m = new MemoryStream();             BinaryFormatter f = new BinaryFormatter();              // Example 1             Func&lt;int&gt; succeeds = () =&gt; 5;             f.Serialize(m, succeeds);              // Example 2             Other o = new Other();             Func&lt;int&gt; fails = () =&gt; o.Value;             f.Serialize(m, fails); // throws SerializationException - Type 'Example.Program+&lt;&gt;c__DisplayClass3' in Assembly 'Example, Version=1.0.0.0, Culture=neutral, PublicKeyToken=null' is not marked as serializable.         }     }   Listing 2  This is similar to the issue of trying to serialize iterators and I had found the following code in a previous search (see ) Using the code from Listing 1 and an ISurrogateSelector I was able to successfully serialize and deserialize the second failing example.  Objective  I have a system that is exposed via a web service. The system has a complex but small state (many objects, not a lot of properties per object). The state is saved in the ASP.NET Cache, but is also serialized to a BLOB in SQL in case of cache expiration. Some objects need to execute arbitrary "events" upon reaching some condition. Hence they have properties accepting Action/Func objects. Contrived example:      class Command     {         public Command(Action action, Func&lt;bool&gt; condition);     }   Somewhere else      void DoSomethingWithThing(Thing thing)     {         state = Store.GetCurrentState();          Command cmd = new Command(() =&gt; thing.Foo(), () =&gt; thing.IsReady())         state.Add(cmd);          Store.Save(state);     } Add warning when changing display name The policy on changing one's display name :  only one display name change is allowed every 30 days  However, there is no warning whatsoever when actually changing one's display name.  Thus, it came as quite a bit of a shock to me when I found out today that I might be stuck as &quot;Unicorns are Yummy&quot; on Stack Overflow for the next freakin' month. I would like to recommend that a warning be added to the &quot;Edit Profile&quot; page to warn people. Or, at least make the policy concerning display name changes more obvious. Workaround I was able to change my display name back (phew!) by copying my profile from Meta Stack Overflow. I still think a warning should be implemented, though. Balloons and lifting gases Hydrogen is the lightest element, so it's cable of lifting the most weight in out atmosphere (probably not the best terminology there, but you get the picture)  Would hot hydrogen (in the same sense as hot air) be able to lift even more mass? Would a higher or lower density of hydrogen in a ballon lift more? If you could have a balloon which had nothing in it (it was a vacuum inside) would that lift more than a hydrogen balloon?  Basically what are the physics to balloons and lifting?  (really not sure what to tag this, so if someone else could that'd be great) Integral inequality with a strange condition Let $f$ be a continuously differentiable real valued function on $[0,1]$.  It is given that $\displaystyle \int_{\frac{1}{3}}^{\frac{2}{3}}f(x) dx=0$     Find the minimum value of $\dfrac{\int_{0}^{1} (f'(x))^2 dx}{\left( \int_{0}^{1} f(x) dx \right)^2}$   I tried to use Cauchy-Schwarz to show that  $$\frac{\int_{0}^{1} (f'(x))^2 dx}{\left( \int_0^1 f(x) dx \right)^2} \ge \frac{\left( \int_0^1 \bigl| f(x)f'(x) \bigr| dx \right)^2}{ \left( \int_0^1 f(x) dx \right)^2} \ge \frac{ f(1)^2 - f(0)^2}{2 \left( \int_0^1 f^2 (x) dx \right)^2}$$ But I can't proceed from here.  Also, I don't know how to use the condition  $\int_{1/3}^{2/3}f(x) dx=0$ Proving that $\mathbb{Z}[\sqrt{2}]$ is a Euclidean domain We're proving that $\mathbb{Z}[\sqrt{2}]$ is a Euclidean domain, using the norm function $$\nu (a + b\sqrt{2} ) = |a^2 - 2b^2|$$ and the first part says that since $\nu (a + b\sqrt{2} ) = |(a + b\sqrt{2})(a - b\sqrt{2})|$ it's clear that $\nu (xy) = \nu(x) \nu(y)$? ... Can someone please explain to me how this is clear? How to encrypt files using a cross-platform solution? The definite and zero article with the word "civilization" How do you secure wood framing to a concrete basement floor? Let $A$ be an $n \times n$ matrix with real enteries. Define $\langle x,y\rangle_A = \langle Ax,Ay\rangle, \quad x,y \in \mathbb R^n$ , where $\langle,\rangle$ is a standard inner product on $\mathbb R^n$. Then  $\langle x,y\rangle_A$ is an inner product iff  1)  $\ker A = \{0\}$.  2) $\operatorname{rank} A= n$.  3) all eigenvalues of $A$ are positive.  4) all eigen value of $A$ are non negative.  Clearly if $\ker A \neq \{0\}$, then there exist $x \neq 0$ such that $Ax =0$, then $\langle Ax, Ax\rangle_A = 0$, then $\langle ,\rangle_A$ is not inner product, so $\ker A = \{0\}$, thus rank of $A$ is $n$.  I am unable to check the 3) and 4) option. Please help to check the other option. Thank you How to install Ubuntu 18.04 alongside Windows 10 Why does chown report "Operation not permitted" on OS X? I am trying to do the following on my Mac (10.6.7):  sudo chown myusername:wheel ./entries   but Unix/Mac is returning "Operation not permitted". When I ls -lash the culprit file, it looks as follows:  8 -rwxrwxrwx   1 myusername  staff   394B Apr 26 23:26 entries   I've tried sudo and sudo su; nothing works. Any ideas what's up?  I'm trying to chmod files I've copied from my old Ubuntu box. Most of the files have successfully chmod'ed recursively; just this one is stuck and I don't understand why. My multiple regression analysis model has a statistically significant F value however all beta values are statistically non-significant.   All the regression assumptions are met. No multicollinearity was found. Correlations among all predictors are all less than 0.60.   What else might be the cause of the insignificant predictors? How to enable MySQL client auto re-connect with MySQLdb? I came across PHP way of doing the trick:  my_bool reconnect = 1; mysql_options(&amp;mysql, MYSQL_OPT_RECONNECT, &amp;reconnect);   but no luck with MySQLdb (python-mysql).  Can anybody please give a clue? Thanks.