Let $f(x)$ be differentiable in $x \in (0,\infty)$ and suppose $$\lim_{x \to \infty } f(x) + f'(x)  = L$$ , where $L$ is finite quantity then find $$\lim_{x \to \infty } f(x)  \text{ and } \lim_{x \to \infty } f'(x)$$ or can nothing definitive be said about these limits.   Intuitively it seems that $\lim_{x \to \infty } f'(x) = 0 $ because if it is not , as we approach $\infty$ , its value will keep on changing as we go farther towards right. And thus $\lim_{x \to \infty } f(x) = L $. But how to prove that this happens or if it is really this that happens.   Eg like in the function $x\sin \frac{1}{x}$, the limit approaches $0$ but it's derivative keeps oscillating. So maybe , something similar might happen at $\infty$ where the function approaches a limit but it's derivative keeps oscillating.  How do we solve it ?? Please help   Edit: As it has been pointed out that this is a possible duplicate, I must say, that the  does not have a satisfactory answer and consider the follow up argument-  If we consider the function $\frac {e^x f(x) }{e^x} $ And apply L' Hopital's Rule, as @ΜάρκοςΚαραμέρης points out Even in the case if the limit of $f(x)$ exists consider for example $f(x)=e^{-x}sin(\frac{1}{x})$, how is L Hospital applicable now? You need a formal proof that $\lim_{x\to\infty}e^xf(x)$ exists  Also as @PeterForeman point out, in the accepted answer it is assumed that $f'(x) \to 0$, instead of proving it.
I want to prove or disprove this problem: If there exist $\lim\limits_{x\rightarrow \infty} (f'(x)+f(x))=L&lt;\infty$ then $\lim\limits_{x\rightarrow\infty} f(x) =L$.  When I assume problem below:  If there exist $\lim\limits_{x\rightarrow\infty} (f'(x)+f(x)) =L&lt;\infty$, There exists $\lim\limits_{x\rightarrow\infty} f(x)$?  I can use mean-value theorem to show that.  So my question is:     If $\lim\limits_{x\rightarrow\infty} (f'(x)+f(x))=L&lt;\infty$, does $\lim\limits_{x\rightarrow\infty} f(x)$ exist?
I am trying to read a custom (non-standard) CSS property, set in a stylesheet (not the inline style attribute) and get its value. Take this CSS for example:  #someElement {   foo: 'bar'; }   I have managed to get its value with the currentStyle property in IE7:  var element = document.getElementById('someElement'); var val = element.currentStyle.foo;   But currentStyle is MS-specific. So I tried getComputedStyle() in Firefox 3 and Safari 3:  var val = getComputedStyle(element,null).foo;   ...and it returns undefined. Does anyone know a cross-browser way of retreiving a custom CSS property value?  (As you might have noticed, this isn't valid CSS. But it should work as long as the value follows the correct syntax. A better property name would be "-myNameSpace-foo" or something.) I am trying to create the following trigger:      CREATE TRIGGER update_customers AFTER INSERT ON wp_usermeta     FOR EACH ROW      BEGIN             if new.wp_metakey = first_name then insert into test_customer(first_name) values (new.wp_meta_value);     END   I keep getting a 1064 error that says, "You have an error in your SQL syntax; check the manual that corresopons to your MySQL server version for the right syntaxt to use near " at line 5"  What am I doing wrong? Citrix receiver error 1000119 I am trying to run Citrix receiver on the Xenial version of Ubuntu, but I am getting this error below  ERROR: Cannot connect to &quot;0.0.0.2 - Remote Desktop Connection&quot; Unknown error 1000119. Verify your connection settings and try again.  What should I do? Difference Between Sequential Workflow & State machine workflow? What is the difference between Sequential Workflow and Stater machine workflow? What context should I take account to when choosing one of them? Do we have negative prime numbers?   $..., -7, -5, -3, -2, ...$ Are all simple left modules over a simple left artinian ring isomorphic? Selecting features that do NOT intersect in ArcGIS Desktop? Etymology of Tor and Ext Functors The names of the derived functors $\operatorname{Tor}$ and $\operatorname{Ext}$ seem quite cryptic to me. Does anyone know what these abbreviations stand for? I would be glad if someone could tell me where these names come from. What lubricants can be safely added to a laptop fan? I've got a fan in a laptop that occasionally makes some loud airplane-like noises.  It mostly happens about 10 seconds after booting up and lasts for 10-20 seconds.  It also occasionally happens when the computer is running, but very rarely.  Interestingly, it does not seem to happen when the processor is being used heavily.  The laptop does not have a discrete graphics card.  The fan is clean.  I'm not sure if the laptop is under warranty, but I trust my own skills much more than I trust the manufacturer (or the shipper).  Before I search for a matching fan and replace it, I thought I would see if it is safe to try any lubricants.  Are there any lubricants that are safe to use inside a laptop?  I don't want a fire or anything that will splatter and cause conductivity issues within the circuitry. I'm more of a math guy than a physics guy so bear with me....  In fractal geometry, fractals are considered to have fractional dimension. For instance an object such as the Koch curve has a fractal dimension of around $1.2$. You can also define a concept called fractal extent (which is the experimental version of hausdorff measure). This extent is a measure of how large a fractal is. It's even been used to more accurately model the distribution of coastline bird nesting.   Even though fractals work as an interpretation of geometrical units, I don't think they would describe things such as $kg^{1/2}$ very well. In addition, I've heard about white Gaussian noise that has square root time units.   What is the physical meaning of fractional units such as the above? (Since it obviously has little geometric meaning) Can blender use the Alpha channel of an image as transparency on a plane? What is the right way to indent your source code? I have been having a discussion regarding the Shape Water cantrip from the Elemental Evil Player's Companion. A friend and I are trying to come to an accurate view of how this spell actually functions.   Some example questions:   Must the targeted water be within a 5 foot cube area, or does it need to be of a volume not exceeding 125 cubic feet? If you shape water that is within a 5 foot cube, can the final shape you animate it to extend outside the cube? Can you cause the shaping to affect ice and snow, or only liquid water? Would you be able to carry a globe of water alongside you over distances?   Really, there are two parts:    What can be targeted by the spell? What are the limitations on the 'shape and animate' portion of the spell?   The answer I am looking for, if possible, would use the language of the spell, possibly with precedent from other game text, to clearly define the limitations of this cantrip. Conceptual half-life Question So I have this word problem and I’m a bit confused about it. I have the answer and explanation but I still don’t understand: The half-life of carbon-14 is approximately 5730 years, while the half-life of carbon-12 is essentially infinite. If the ratio of carbon-14 to carbon-12 in a certain sample is 25% less than the normal ratio in nature, how old is the sample?     A. Less than 5730 years      B. Approximately 5730 years      C. Significantly greater than 5730 years, but less than 11460 years      D. Approximately 11460 years  Correct Answer: A Explanation: Because the half-life of carbon-12 is essentially infinite, a 25 percent decrease in the ratio of carbon-14 to carbon-12 means the same as a 25 percent decrease in the amount of carbon-14. If less than half of the carbon-14 has deteriorated, then less than one half-life has elapsed. Therefore, the sample is less than 5730 years old. Be careful with the wording here—the question states that the ratio is 25% less than the ratio in nature, not 25% of the ratio in nature, which would correspond to choice (D). How is the ratio of the carbon isotopes relevant to half-lives? What is the purpose of saying the half-life of an isotope is infinite? What is meant by “the normal ratio in nature”? Just based on the answer, it seems like the question said “25% of a carbon sample decayed, how old is this sample?” Obviously it’s younger than 5,730 years (the time for its first half life) because if only 25% decayed that means half a half-life has passed. I don’t see how isotopes and ratios would change this problem at all. Edit: Every answer here was very informative and helpful. It was tough to pick a best answer. I picked Cosma’s because it clicked with me. If you’re reading this and want to understand radiocarbon dating, look at every answer because there is some useful information in every answer How do I make an object face the mouse position, but only on one plane? I want a game object to turn and look towards where the mouse was clicked (in world space).  It works well when considering all 3 dimensions, but I really only care for the plane formed by the x and z axis, not the y.  I've tried to fix this by setting the y of the directionTarget to the y of the object's position, but that gives weird results.  My code looks like this:  Quaternion rotationTarget;     float rotationLerpProgress;     float rotationLerpDuration = 1f;      // Use this for initialization     void Start ()     {      }      // Update is called once per frame     void Update ()     {             if (Input.GetMouseButtonDown (0)) {                     var ray = Camera.main.ScreenPointToRay (Input.mousePosition);                     RaycastHit rayHit;                      if (Physics.Raycast (ray, out rayHit)) {                             var rayHitPoint = rayHit.point;                              var rotationDirection = transform.position - rayHitPoint;                                                            rotationDirection.Normalize ();                             //rotationDirection.y = transform.position.y;                             rotationTarget = Quaternion.LookRotation (rotationDirection);                             rotationLerpProgress = Time.deltaTime;                     }                     //targetPosition = Camera.main.ScreenToWorldPoint (Input.mousePosition);                                     }              if (rotationTarget != transform.rotation) {                      transform.rotation = Quaternion.Lerp (transform.rotation, rotationTarget, rotationLerpProgress / rotationLerpDuration);                     rotationLerpProgress += Time.deltaTime;                       if (rotationLerpProgress &gt;= rotationLerpDuration) {                              transform.rotation = rotationTarget;                     }              }      }   I must be missing something. How do I do this right? Simplest way to send one-line mail out via command line using gmail? Using Raspbian and Ubunntu 16.04 LTS so need a generic Linux solution.  Requirement is simple:  I need a way to send one-line email messages from the command line.  I have set up a gmail account just for this particular Rpi3, with the address of rpi3abc@gmail.com - with no 2FA  So now I need to be able to send one-line mail messages from anywhere (including cron) without user intervention.    I also would like it to be able to send text files; basically, anything from stdin. Convert a Cannondale SM1000 (about 1991 model) to disk brakes I've got an older, about 1991, model Cannondale SM1000 mountain bike purchased new.  Would it be worth it to convert to disk brakes and if so what would be the process? How quickly is the desired state of curl of the Electric Field around a point achieved where a Magnetic Field Vector has just begun changing? My question and context with explanation are given below. Thank you in advance.   Faraday’s Law () states that a time-changing magnetic field vector induces a curl of the Electric field around that point. However it does not specify how quickly the necessary spatial gradient is developed in the Electric Field to satisfy that curl. Of course, the Electric Field vectors around that point cannot just jump to the required values instantaneously in accompaniment as soon as the magnetic field starts changing at a fixed rate, since it is required that the Electric Field has a sensible rate of change to in turn induce a curl of the magnetic Field and continue the propagation of the wave. It would seem the exact rates at which the Electric and Magnetic Fields are changing is critical in the mechanism of an EM wave’s propagation. How would I then calculate the rate of change of E somewhere along the wave given I know the rate of change of the B vector that began it, and given the wave is restricted to one dimension for simplicity? There should after all be some delay in how the EM wave moves from point to point in space, seeing as it moves at the finite speed of causality (3*10^8 m/s), as shown in the first image:    Some of my ideas so far have been that perhaps the rate of change of E is determined such that it exactly cancels the growth of the time-changing E vector that gave rise to a curl of B before, which developed at such a rate to induce a curl of E, which demands this rate of change of E which I am talking about now. In this way the wave might level out the growth of the fields in its wake and spread the change further outwards, in the same way smoothening out a crease spreads it elsewhere. However I am stuck on how to show this rigorously.  I am open to any difficulty of maths, so long as the answer provides the explanation. Below are some diagrams to help visualize what I am talking about:    Faraday's Law requires that a curl of E is developed, which under these restrictions requires a spatial gradient of E to appear across that point. Since the field across that point was previously in some other state, it is required that the Electric Field vectors that are infinitesimally close to that point undergo a rate of change through time to achieve the desired state of curl. The question is, how quickly is this desired state of curl achieved? How to know current name of the database in Django? Why and how do "bigger pixels" make a better picture? This is going off of today's iPhone 5S announcement, when Phil Schiller said that "bigger pixels" is the key to a better picture.      What does this mean? Is it true or just a marketing ploy?