Does the following limit exist? $$\lim_{x \to 0} \frac{1}{x}\sin\left(\frac{1}{x}\right)$$  If yes, then I have to find it; if no, then I need to give reason why. I couldn’t figure out how to prove this formally, can someone please help me out. Edit: please note that sin is multiplied by 1/x here and not x.
Does the following limit exist? $$\lim_{x \to 0} x\sin\left(\frac{1}{x}\right)$$  If yes, then I have to find it; if no, then I need to give reason why. I couldn’t figure out how to prove this formally, can someone please help me out.
Does "corresponding author" carry an implied meaning? how to Query parent to child I have process in salesforce in such a way opportunity we can relate this opportunity to this lookup field parent_opportunity__c . Which relates opportunity to other parent opportunity.  I want to write a query parent to child :  select id,(select id from opportunity where parent_opportunity__c!=null) from opportunity where parent_opportunity__c=null   I have worked with different scenarios i cant able to crack it  select id,name,(select id,name from parent_opportunity__r) from opportunity this is showing error and   select id,name,(select id,name from opportunity) from parent_opportunity__c this is also error Understanding MongoDB BSON Document size limit The Complete Rate-Limiting Guide In the book I'm reading at the moment () it says that if you don't declare a default constructor for a class, the compiler supplies one for you, which "zeroes out each data member". I've experimented with this, and I'm not seeing any zeroing -out behaviour. I also can't find anything that mentions this on Google. Is this just an error or a quirk of a specific compiler? Every cyclic group with order > 2 has at least 2 distinct generators Every cyclic group with order > 2 has at least 2 distinct generators   Here's what I've got so far:  Either order of our group, $G$, is finite or infinite.  Suppose infinite: then our group is isomorphic to $\mathbb Z$ under addition. This group has two distinct generators, therefore so does $G$.  Suppose finite: Then $G$ is isomorphic to $\mathbb Z_n$ under addition modulo $n$.  We know order of $G$ is $\geqslant 3$. If order of $G$ is odd, both 1 and 2 are generators of $\mathbb Z_n$ under addition modulo $n$, so $G$ has at least two generators.  If order of $G$ is even, then both 1 and some odd number between 1 and $n$ are generators. How do we determine this odd number?  Does this look like I'm on the right track? How do I get Siri to pronounce my name correctly? Siri doesn't pronounce my name correctly. I tried editing the phonetic spelling as it says on many sites, but it doesn't fix the problem. How do I fix this problem? Delete files except n files with the "bigger" name? Some kind of + (singular or plural)? I have a question about using the phrase "some kind of _".  Is it incorrect to say "some kind of + plural noun"?     For example, is it wrong to say "are these rock-like things some kind of chocolates?"   I didn't use "kinds" and that kind of adds to my question. Is "kind" always plural, like "some kinds of chocolates"? Because although I'm using "some", it feels like "kind" shouldn't be a countable noun.  Anyway, I don't think I've heard of "some kind of + plural", but I think it makes more sense then using singular, since I think chocolates are countable. Please help! These two exercises I encountered recently seem to develop some type of connection between GCD and LCM I can't quite figure out. Exercise 1:  Find all the numbers $x$ and $y$ such that: $a) \ GCD(x,y)=15, \ LCM(x,y)=150$ $b) \ GCD(x,y)=120 \ LCM(x,y)=1320$ $c) \ GCD(x,y)=100 \ LCM(x,y)=990$  Exercise 2:   Find all the numbers $m,n$ such that $GCD(m,n)=pq , \ LCM(m,n)=p^2qs$  where $p,q,s$ are prime  The first thing that is known to me is that $GCD(x,y) \cdot LCM(x,y)= x \cdot y$ Also $LCM(x,y)$ is at most $x \cdot y$ while $GCD(x,y)$ is at most $\max \{x,y\}$. Last thing is that $GCD(x,y)|LCM(x,y)$. Using all this I tried to solve the first exercise: $a)$ First two obvious pairs are $x=15, y=150$ and $y=15, x=150$. Now neither of the numbers can be bigger than $150$ or smaller than $15$. So we are looking for numbers in the range $15-150$ that satisfy $x \cdot y = 15 \cdot 150$ Another such pair is $(x,y)=(30,75), \ (x,y)=(75,30)$. Similarly for $b)$ we find that the only possible values are permutations of the set {$120,1320$} and in $c)$ since $100$ does not divide $990$ no such numbers exist. Now exercise 2 is what made me think there is actually another connection I'm not quite aware of since now it's about arbitrary prime numbers and the previous method doesn't work anymore. My intuition is that it has something to do with $GCD$ or $LCM$ of the $GCD(x,y), \ LCM(x,y)$ How to make tr aware of non-ascii(unicode) characters? how do I print an unsigned char as hex in c++ using ostream? I want to work with unsigned 8-bit variables in C++.  Either unsigned char or uint8_t do the trick as far as the arithmetic is concerned (which is expected, since AFAIK uint8_t is just an alias for unsigned char, or so the debugger presents it.  The problem is that if I print out the variables using ostream in C++ it treats it as char.  If I have:  unsigned char a = 0; unsigned char b = 0xff; cout &lt;&lt; "a is " &lt;&lt; hex &lt;&lt; a &lt;&lt;"; b is " &lt;&lt; hex &lt;&lt; b &lt;&lt; endl;   then the output is:  a is ^@; b is 377   instead of   a is 0; b is ff   I tried using uint8_t, but as I mentioned before, that's typedef'ed to unsigned char, so it does the same.  How can I print my variables correctly?  Edit: I do this in many places throughout my code.  Is there any way I can do this without casting to int each time I want to print? This is about a step in a derivation of the expression for the relativistic Doppler effect.  Consider a source receding from an observer at a velocity $v$ along the line joining the two. Light is emitted at frequency $f_s$ and wavelength $\lambda_s$. The frequency $f_0$ and wavelength $\lambda_0$ received by the observer will be different.  Some textbooks now argue that the wavelength received by the observer is given by $\lambda_0=(c+u)T_0$, where $T_0$ is the time period of the wave in the observers frame. The argument given is that the successive "crests" will be an extra distance $uT_0$ apart due to movement of the source. Upon relativistically transforming the time period in the source frame to the observer frame, the correct result $f_0=f_s\sqrt{\frac{c-u}{c+u}}$ is obtained.  But the argument for the wavelength seems reminiscent of the classical Doppler effect for sound. Is it really applicable here through this argument? And is there a way to show the same result for $\lambda_0$ mathematically? How to add custom Modules in magento Entire function and the series of its successive derivatives I recently came across the result that if $g$ is holomorphic in a neighbourhood of a point $z_0$ and  $$\sum_{n=0}^{\infty }{g^{(n)}(z_0)} $$ converges then $g $ is an entire function. Now, my doubt is if the above series is absolutely convergent then can we say something more about $g$, like say $g$ is a polynomial? I could not find any non-polynomial  entire function( I tried with functions like $e^z$, $\sin z $, et al), whose successive derivatives converge absolutely as series. Any help please? I can't seem to find info on installing a Swedish language package on my machine. The language is not listed  and the hits on Google I got led nowhere (or, rather, to the said page).  Where can I ask such a question (within the realm of Stack Exchange)? How to answer a negative question in English? Q: "Don't you know?"   If I really don't know, how should I respond to this question? Yes or No?  There's a similar question on EL&amp;U, but the answer isn't clear how it works:      Also here on ELU too:      These answers are confusing. I want to know what one word answer I can use. In other words I want to know:   What will native speakers understand if I reply No to this question? What will native speakers understand if I reply Yes to this question? Are there any stats about the percentage of close reasons used to close questions   Successfully (preferably, based on the number of closed questions) Unsuccessfully (alternatively, the total percentags of close votes cast by users)   ?   Is there any way to get this from the query explorer? Would someone explain wheel sizes.  If I am not mistaken, the wheels on my Trek 520 touring bike are 700C. There are no markings, and of course, no documentation of any kind. I assume 700 means 700mm, so 700 / 25.4 = 27.56 inches.  If I measure the rim however, its diameter is approximately 25.1 inches.  If I buy Continental tires, the size is 28 inch, but I have to buy 27 inch Bontrager tubes.  To make matters worse, per this link, tires come in sizes of 26, 27, and 28 inch, and size 700.     How can I determine my wheel size, and are these various sizes interchangable. Also, is the 'C' suffix important, and what does it mean?  Of course the tires are marked, but I am under the impression (per the dealers comments) that Continental marks their tires differently than everyone else. Also, the tire markings don't help for ordering a new wheel.   With so many tire sizes available, how can I know if I am using the correct size.  After reading the various answers here, I should explain that my confusion on this matter stemmed from the fact that I used to work as an auto mechanic. In the auto industry, car tires are based on wheel size, 13", 14", 15" etc, and there is no possible way to mount a 15" tire on a 13" rim for example.  So when I saw all these different tire sizes, I assumed they all required a different wheel size, but in fact, there is essentially just one wheel diameter for my type of bike, which they refer to as 700C, which is actually 622mm.   I also initially assumed that the C referred to the wheel's width, but that isn't true either. The C is apparently just antiquated terminology.  So at least in theory, I can mount a 26", 27" 28" or 29" tire on my 700C (622mm) rim. I just need to verify the width of the tire is suitable my wheel width, which is what Sheldon Brown's article explains.  Also, I measured my Continental tires, which are marked as 28 inch. They actually measure 27 inches (fully inflated), which must be why my dealer calls their sizing non standard. How to determine exact contrast ratio Because of the recent Web accessibility Law in my region, I need to ensure I'm designing with an appropriate contrast ratio.  WCAG says the value of this ratio is 4.5:1    How do I figure this out in in Photoshop and Illustrator?