Prove the following using epsilon-delta definition:  Let $a \in \mathbb{R}$ and $f,g$ be defined on $\mathbb{R}$.  IF $\lim_{x\to a} f(x) = 647$ AND $\lim_{x\to a} g(x) = \infty$, then  $$\lim_{x\to a} f(x) + g(x) = \infty.$$  Not sure about this... I've set up the definitions. I believe we will have to take a minimum between the deltas for first two limits then use it in the third.  Anyone have any clues?
Prove the following using epsilon-delta definition:  Let $a \in \mathbb{R}$ and $f,g$ be defined on $\mathbb{R}$.  If  $$\lim_{x\to a} f(x) = 647 \tag 1$$  and $$\lim_{x\to a} g(x) = \infty \tag 2$$ then   $$\lim_{x\to a} f(x) + g(x) = \infty.$$  $$(1) \implies\forall \epsilon &gt; 0, \exists \delta &gt; 0, s.t. |x-a|&lt;\delta \implies |f(x)-647| &lt; \epsilon$$  $$(2) \implies \forall M &gt; 0, \exists \delta_2 &gt; 0, s.t. |x-a|&lt;\delta_2 \implies |g(x)| &gt; M$$  Not sure about this... I've set up the definitions. I believe we will have to take a minimum between the deltas for first two limits then use it in the third.  Anyone have any clues?
Clone Method not working I am trying to Clone a lead when certain condition is arrived using before trigger but i am not able to see cloned record.   for( Lead ld : Trigger.new ) {                          if(ld.Integration_Source__c == 'Hubspot' &amp;&amp; Trigger.isUpdate){         Lead LeadBeforeUpd = Trigger.oldMap.get( ld.Id );         if(ld.Program__c != LeadBeforeUpd.Program__c ){            Lead ClonedLead = new Lead();            ClonedLead = LeadBeforeUpd.clone(false, false, false, false);         }     }  } How do I create this crease effect on glass? Why can't Chromium suddenly access any partition except for /home? Changed response to Recommendation questions now that there's a site for them? I am looking to replicate black &amp; white maps in QGis, using hatching patterns and greyscales to indicate different categories of a continuous variable (example below).  It is possible/straightforward to do so in QGis? At first sight I only see color gradients and symbol fills.   ). Do full frame sensors gather more light than crop sensors? Lots of people say that full frame sensors receive more light than cropped sensors. I have never found a proof of this claim so I tried to do the computation by myself, and proved the contrary! Could you tell me if I am wrong ?  We want to compare the same frame with the same depth of field. I'm not concerned by the quantity of photons by photosite, which is unrelated to sensor size but to the density of pixels. I have neglected effects like vignetting or angle effect on the edge of the micro-lens array. Here is my simple reasoning :  If you want the same  α~2arctan(size/2f) with a full-frame sensor and a crop sensor with crop ratio c, you have to multiply the focal length by approximately c. Now, in order to maintain the , the f-number N have to be divided by c. If we measure the "amount of light" with the well defined  Ev provided by the same frame of the same scene (so the  is fixed), we have Ev ~ f/N.  Putting all together, Ev_crop = Ev_ff x c², so the cropped sensor receive more light than the full frame sensor !  For those who are interested in the price of two equivalent systems, one with a FF+50mm+135mm and the other with Crop+35mm+85mm, see this . Is there a way to consistently increase the vertical spacing between equation lines in a gather (or gather*) environment from amsmath?  Specifically, I want to produce something like  \begin{gather*} a = b \\[2ex] c = d \\[2ex] e = f \end{gather*}   without having to write [2ex] at the end of each equation line.  I can think of some hacks that redefine \\ to automatically add the spacing, but is there some cleaner/better way to achieve this? On this link (Sentinel1  for a SLC product), the resolution is written as a range from the highest corresponding to low incidence angle, to the lowest resolution corresponding to high incidence angle.  My question is how is the resolution related in general to the incidence angle? And if they are really related, why is the resolution affected by the change in the incidence angle? How to disable Spotify indicator in the top panel? The Spotify indicator icon clashes with the native Unity theme, and is redundant, since it is already listed in the Sound indicator menu. Is there a way to remove or disable the Spotify indicator? Or at least to change the icon (couldn't grep it on my filesystem). I am using the Spotify "preview" with 12.04. Thanks! Reprojecting raster from lat/lon to UTM in R? i have to turn it into a  UTM in order to make the buffer functional.  wets&lt;-readOGR(dsn=".",layer="shapefile") r.raster &lt;- raster() extent(r.raster) &lt;- extent(wets) res(r.raster) &lt;- 100   wets.r &lt;- rasterize(wet,r.raster) plot(wets.r) wetsbuf&lt;-buffer(wets.r,width=500)   During the buffer creation which is the last line of code, it gives this warning:  Warning message:   In couldBeLonLat(x) :   raster has a longitude/latitude CRS, but coordinates do not match that   here's the info    summary(wets.r)           layer  Min.        1  1st Qu.     1  Median      2  3rd Qu.     9  Max.       11  NA's    52629  summary(wets)    Object of class SpatialPolygonsDataFrame Coordinates:       min       max  x  683705  714088.8  y 4326266 4343768.0  Is projected: TRUE   proj4string :  [+proj=tmerc +lat_0=0 +lon_0=24 +k=0.9996 +x_0=500000 +y_0=0 +datum=GGRS87  +units=m +no_defs +ellps=GRS80 +towgs84=-199.87,74.79,246.62]  Data attributes:    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.      0.0     2.5     5.0     5.0     7.5    10.0         wets.r  class       : RasterLayer  dimensions  : 175, 304, 53200  (nrow, ncol, ncell) resolution  : 100, 100  (x, y) extent      : 683705, 714105, 4326268, 4343768  (xmin, xmax, ymin, ymax) coord. ref. : +proj=longlat +datum=WGS84 +ellps=WGS84 +towgs84=0,0,0  data source : in memory names       : layer  values      : 1, 11  (min, max) attributes  :    ID FID  from:  1   0  to  : 11  10   I have to change the prjection in order to be possible to do the buffer. Show how if Alice uses the same value of $k$ to sign two different messages $m_1$ and $m_2$, using the ElGamal signature scheme, Eve can recover the value of $a$ from the corresponding signatures $(m_1, r_1, s_1)$ and $(m_2, r_2, s_2)$. (Note: you are allowed to assume that if $\gcd(a, n) = d$ then there are $d$ solutions to the congruence $ax \equiv b \pmod n$.)     ElGamal Signature Scheme:      Key Gen:      Compute $y = g^x \;\bmod p$.      The public key is $(p, g, y)$. The secret key is $(p, g, x)$      Signature Gen:      Choose a random $k$ such that $0 &lt; k &lt; p − 1$ and $\gcd(k, p − 1) = 1$      Compute $r = g^k \;\bmod p$ and $s = k^{-1}(m – xr) \;\bmod{p-1}$   Thoughts: So far, I can tell that $r_1$ and $r_2$ are equal and $s_1$ and $s_2$ are closely related since the only variation is $m$.  We can relate the two equations for $s$, by solving them for $-xr$:  $$s_1k - m_1 \equiv s_2k - m_2 \pmod p$$  $$(s_1 - s_2)k \equiv m_1 - m_2 \pmod p$$  Let $a = s_1 - s_2$, then we know from the question that for $\gcd(a, n) = d$ there are $d$ solutions for $k$.  Now the forger can compute $g^k$ for each solution to k, until $r$ is found.  Then compute $xr \equiv m_1 - ks_1 \pmod{p-1}$  There are $d&#39; = \gcd(r, p-1)$ solutions for $x$.  The forger can compute $g^x$ for all the $x$s found until she finds $y$.  Once she has $y$ she knows the proper $x$ which can be used to find $m_1$ and $m_2$.  How does this look? Programmatically download (seamless) NED tiles? I used to download 1/3 arcsecond elevation geotiffs from extract.cr.usgs.gov using web services.  But a few days ago, that stopped working, probably because USGS reorganized their site.  I tried emailing USGS and haven't heard back yet.    I could, of course, manually download states or degree tiles using the seamless server, but I want to do it programmatically so that users of my trip planner can get elevation data for their region without human intervention.  And SRTM isn't high enough resolution for bicycle/wheelchair trip planning.  Any ideas? I will be getting a point-and-shoot camera for taking portraits and only portraits.  Specifically portraits with a shallow depth of field.   Groups portraits sometimes.  It's just too much work to add lens blur at post-prod.  I understand that setting a P&amp;S on macro mode gives a shallow focus and DOF.    How do I read P&amp;S camera specs to shortlist those with the shallowest depth of field?   Added: By point-and-shoot, I mean a camera that is easy to use -- Not necessarily compact, inexpensive, or current.  Also, if there are two or three factors/specs to get a shallow depth of field, then which spec contributes the most to bokeh? which is second most contributing? the third, etc.? What is the formula for this numerical series? In the recent post , Werner helped me to redefine a proof environment which can be hidden by an optional boolean argument (showProof). Without an argument, somehow the line/paragraph breaking is not correct. How can one fix this?  \documentclass{scrartcl}  \usepackage{ifthen} \usepackage{environ} \usepackage{amsthm}  \newboolean{showProof}% for showing proofs \setboolean{showProof}{true}% setting the switch \newif\ifstarttheorem  % proof environment (with hide feature) \makeatletter \providecommand{\env@proof@save@env}{} \providecommand{\env@proof@process}{} \RenewEnviron{proof}[2][showProof]{   \gdef\@tmp{#1}%   \ifthenelse{\boolean{#1}}{\par\pushQED{\qed}%     \normalfont\topsep2\p@\@plus2\p@\relax     \trivlist\item[\hskip\labelsep\sffamily\bfseries Proof~#2]\gdef\mycurrenvir{proof}\global\starttheoremtrue\mbox{}\hfill\\*\ignorespaces     \BODY}{\ignorespaces} }[\ifthenelse{\boolean{\@tmp}}{%   \gdef\mycurrenvir{\relax}   \popQED\endtrivlist\@endpefalse }{\ignorespacesafterend}] \makeatother  \begin{document} Just some text \begin{proof}   Just some text in a standard proof. As we can see, there is a problem. \end{proof} \begin{proof}[false]   Proof with optional argument set to \texttt{false}. This proof correctly does   not appear. \end{proof} \begin{proof}[showProof]{(Heading)}   Proof with optional argument (\texttt{true}) and second argument ``(Heading)'' \end{proof} \end{document} Ubuntu 18.04 - Dell XPS13 9370 no longer suspends on lid close This was working perfectly on 17.10 but after upgrading to 18.04 yesterday, when the lid is closed the screen turns off but doesn't suspend properly.  I travel around a lot and immediately noticed the heat (and battery draining) when taking it out of the travel case.  I've tried uncommenting these lines in /etc/systemd/logind.conf  HandleLidSwitch=suspend HandleLidSwitchDocked=suspend   and restarted but didn't make any difference. Why does cout print char arrays differently from other arrays? Markov's paper on insolubility of the homeorphy problem I am looking for an English translation of Markov's 1958 paper, On insolubility of the homeorphy problem, which I remember coming across on a website for a computational topology course (taught by Afra Zomorodian). The derivation of the Hamilton-Jacobi equation using canonical transformations is typically done involving a type-2 generating function.  Is it possible to use a another type of generating function, namely type 1, 3 or 4? Why use pointers?