Let $f(x)$ be twice differentiable on $(0,\infty)$ and let $\lim_{x\to \infty} f(x) = L&lt;\infty$ and $|f''(x)| \le M$ for some $M&gt;0$.  Prove that $\lim_{x \to \infty} f'(x) = 0$.  I've tried to use taylor with remainder in a lot of different ways and am still unable to crack this one.
Let $f\colon\mathbb R\to\mathbb R$ be twice differentiable with $f(x)\to 0$ as $x\to\infty$ and $f''$ bounded. Show that $f'(x)\to0$ as $x\to\infty$. (This is inspired by  to a different question)
A linking verb is a verb which connects a subject to its predicate   without expressing an action. A linking verb is used to re-identify or   describe its subject.      Ex:      Alan is a beast.      Alan seems drunk.   But I would say that some linking verbs can be used with "continuous tenses" &amp; other linking verbs can not, but I am not sure.      (linking verb) to experience a particular feeling or emotion      Ex: I’m feeling a little better today.      Do you still feel hungry?   So, feel in this meaning can be used with continuous tenses  But I would think this sentence "Do you still feel hungry?" does not make much sense because it uses "simple present tense" which expresses something that happens regularly. Noone would feel hungry regularly.  So, that sentence "you feel hungry" would mean "you always feel hungry".      (linking verb) (not used in the progressive tenses): to have a   particular physical quality which you become aware of by touching         adj.         The water feels warm.      Its skin feels really smooth.    says "Speakers sometimes use the Simple Present to express the idea that an action is happening or is not happening now. This can only be done with Non-Continuous Verbs and certain Mixed Verbs."  so, "The water feels warm." could mean "The water is always warm" (simple present expressing things that happen regularly - more or less permanent action)  But "The water feels warm." could mean "The water is warm now" (simple present with non-continuous verb expressing things that is happening now - a temporary action)     : (linking verb) to seem; to appear         adj.         to look pale/happy/tired      That book looks interesting.   In dictionary,      (linking verb) (not used in the progressive tenses): to   give the impression of being or doing something         adj.         She didn't appear surprised at all.       (to somebody) (to be) something (not used in the progressive tenses): to give the impression of being or doing something (synonym:   appear)         adj.         You seem happy.   So, we cannot say "you are seeming happy" or "you are appearing happy".  But I am not sure I can say "You are looking good" (sounds right to my ears) or "That book is looking interesting." (doesn't sound right to my ears)  But we can say "The future’s looking good." ()  So, some verbs has the same meaning but one can be used with "continuous" &amp; the other can not, right?  Ex: "you are looking happy", but "you seem happy". Finding $\int \frac {dx}{\sqrt {x^2 + 16}}$ I'm thinking of running a Pathfinder campaign for Skype (text only) and I don't have a group of players. How can I go about recruiting people for play-by-chat games instead of in-person? Are there any online sites in which you can look for people that are looking to play in an online play-by-chat RPG Campaign? Codomain of function complete implies function space complete Let $(X,d_X)$ and $(Y,d_Y)$ be two metric spaces.   Let $(C_b(X,Y), d_C)$ be a function space with the metric $$d_C(f,g):=\sup\limits_{x\in X} d_Y(f(x), g(x)).$$  Prove that if $(Y,d_Y)$ is complete then $(C_b(X,Y), d_C)$ is complete.  Unfortunately, I have no idea how to approach this problem. I can think of something like this:  Let $(y_k)_k\subset Y$ be a Cauchy sequence converging to $y\in Y$. Then there exists some $f\in C_b(X,Y)$ such that $f(x_k)=y_k$ for all $k\in \mathbb N$. Since $f$ is continuous, $\lim\limits_{k\to\infty} d_Y(f(x_k), y)=0$.  But how do I go from here? As far as I understand, what needs to be proved is that any Cauchy sequence of functions in $C_b(X,Y)$ converges in $C_b(X,Y)$, that is, that the limit is bounded and continuous. But how do we devise such a sequence of functions? A particle experiences an acceleration described by  $$ a=kx^{-2} $$ where x is the displacement from the origin and k is an arbitrary constant.  To what value does the velocity v of the particle converge to as x approaches infinity if the particle starts at some point x0?  If I approach this problem with energy, then $$ W = \int F \mathrm{d}x $$ $$  = \int_{x_0}^\infty mkx^{-2} \mathrm{d}x $$ $$  = mk(-\infty^{-1}+{x_0}^{-1}) $$ $$ W = K = mk{x_0}^{-1} $$ $$ \frac{1}{2}mv^2 = mk{x_0}^{-1} $$ $$ v = \sqrt{2k{x_0}^{-1}} $$  How would I solve this problem with pure kinematics? (there appears to be some sort of cyclical dependency where acceleration affects velocity, velocity affects displacement, and displacement affects acceleration)  Likewise, two particles experience accelerations described by $$ a_1=k_1x^{-2} $$ $$and $$ $$ a_2=-k_2x^{-2} $$ where x is the distance between the two particles  What two velocities do the particles reach as x approaches infinity if the two particles are initially separated by some x0? I have the following sequences $(a_n)$, $(b_n)=(a_{2n})$, and $(c_n) = (a_{2n-1})$.  I have to show that:   if $(b_n)$ and $(c_n)$ both convergence to $a$, then $(a_n)$ also converges to $a$; if $(a_n)$ converges then $(b_n)$ and $(c_n)$ also converge; there exist $(a_n)$ not converging but with $(b_n)$ and $(c_n)$ converging.   I am not sure how to solve this since I only know that $(b_n)$ > $(c_n)$ > $(a_n)$ on the terms how the sequences are defined. How should one maintain and update a MiKTeX installation? Every once in a while after updating MiKTeX, the installation gets into an inconsistent state. This might be a problem with the packages (an example is described in ), but sometimes packages are split or renamed. The cases I remember are when  and when miktex-etex-base-2,9 was renamed to miktex-etex-base-2.9 (if I recall correctly; I am sure it was either that or something very similar). In each case the MiKTeX Update process suggested to remove the "old" packages but it didn't give the user any hints about him having to manually add the respective replacement packages back in the package manager to get one's MiKTeX installation back into a usable state.  What are best practices for keeping MiKTeX up to date? Note that I'm not asking about installing new packages or fonts in general; there are other questions covering this area.   In which order should one execute the Update process and the Synchronization of the repository? , it is recommended to check the MiKTeX homepage before updating. What else should one do? When and in which order (and why) should one execute the "Refresh FNDB" and "Update Formats" processes within the MiKTeX Options menu? What else can happen? Other advice? I have tried as following:  1. Turn off: export DISPLAY=:0.0 &amp;&amp; xset dpms force off  2. Turn on: export DISPLAY=:0.0 &amp;&amp; xset -dpms   Which works but that is a temporary solution, which does not fix my problem yet. When i reboot system it start again and again the same auto turn off mode.   How can i now remotely using SSH turn it off forever?  IMPORTANT:  - KIOSK (ATM machine will use this option) - 24/7 advertising display/slides will use this   Optional Note: this guide i followed   , but permanent solution did not worked. Can Mathematica do two evaluations at the same time? Why does nothing show up in the Compositing tab? I am trying to follow Blender Guru's beginner tutorial. He went to the Compositor tab and was using it to Denoise an image but when I got there, no nodes show up. Can you guys help? 'pyrcc5' is not recognized as an internal or external command I installed QGIS 3.0 on Windows 10. I get this error everytime i run the pyrcc4 or pyrcc5 command:  'pyrcc4' is not recognized as an internal or external command  Even if I find a place with a pyrcc4.exe file or a place with apyrcc5.bat file and add the path to the environment variables I get the same error. In the previous installation of osgeo4w i had a pyqtdev folder or something similar appear I think it was on C/. Was there supposed to be such a folder, do I need to download pyqt separately? Using the residue theorem, calculate $\int_{0}^{2\pi}\frac{1}{1-2a\cos{\theta}+a^2}d\theta$ Why Session object destruction failed What should main() return in C and C++? Prove that a non-empty closed convex $S$ has an extreme point iff it does not contain any line I am wondering how to prove the following theorem: Let $S \subset \mathbf{R}^{n}$ be a non-empty closed convex set. Then $S$ has at least one extreme point iff $S$ does not contain any line. Extreme point $x \in S$ means that there is no couple of points $y, z \in S$ and $\lambda \in (0;1)$ such that $x = \lambda y + (1-\lambda)z$. I know that every non-empty closed convex set can be represented as an intersection of all closed half-spaces containing the set but I do not know if it is helpful in proving the theorem. Scoring matrices (BLOSUM & PAM) in BLAST and other sequence-comparison programs The  states that:      The scores are created by comparing the word in the list in step 2* with all the 3-letter words. By using the scoring matrix (substitution matrix) to score the comparison of each residue pair, there are 203 possible match scores for a 3-letter word. For example, the score obtained by comparing PQG with PEG and PQA is 15 and 12, respectively with the BLOSUM62 weighting scheme.   I do not understand how they got a score of 15 for PQG v. PEG. What is a scoring matrix, how is it computed, and how is it used?      Step 2 on that page is Make a k-letter word list of the query sequence: “Take k=3 for example, we list the words of length 3 in the query protein sequence (k is usually 11 for a DNA sequence) sequentially, until the last letter of the query sequence is included”. 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. find the minimum value given constraints Add an SQLite database to an iPhone app scp copy over ssh doesn't work - permission denied error, please?