i'm watching  . and the answers and they're all fine but there is something that bugging me about "rounding down" i attach it below  "for $x &lt; 0$ but sufficiently close to $0$ (e.g., $x = -0.1$), $\lfloor \sin x \rfloor$ is no longer zero, but $-1$, because you are "rounding down."  Therefore, $f(x) = x/(x+1)$"  and unfortunately people who answer this is no longer active , so can someone help me how that $rounding down" work ?
In my assignment I have to prove that the following function doesn't have a limit:  $$f(x)=\frac{x}{x-\lfloor{\sin x}\rfloor}$$ when $${x \to 0}$$  In other words, I have to prove that for every $\delta$ such that $0&lt;|x-{x_0}|&lt;\delta,$ $|f(x)-L|&gt;\epsilon$.  In fact, I don't even know how to start. Do I choose my $\delta$? I know that the sin function has certain values that's supposed to help me. How do I use it?  I have to prove this by definition, so I can't use limit arithmetic for example  Thanks.
PHP Error - Uploading a file Combining jquery functions - on() hover/mouseenter/mouseleave Are spells learned from feats considered to be associated with your class? The multiclassing spellcasting rules (PHB p. 164, or ) state:  Each spell you know and prepare is associated with one of your classes, and you use the spellcasting ability of that class when you cast the spell.  According to the above, every spell you know or prepare should be associated with a class.  However, for spells learned from feats like Drow High Magic or Wood Elf Magic are these actually associated with a class? If so, what class? If not, is the PHB statement above wrong/outdated? Are the spells learned in these feats (and others like it) associated with something else (like your race, in these cases)?  For reference, the  racial feat (XGtE p. 74) states:  You learn more of the magic typical of dark elves. You learn the detect magic spell and can cast it at will, without expending a spell slot. You also learn levitate and dispel magic, each of which you can cast once without expending a spell slot. You regain the ability to cast those two spells in this way when you finish a long rest. Charisma is your spellcasting ability for all three spells.  The  racial feat (XGtE p. 75) states:  You learn the magic of the primeval woods, which are revered and protected by your people. You learn one druid cantrip of your choice. You also learn the longstrider and pass without trace spells, each of which you can cast once without expending a spell slot. You regain the ability to cast these two spells in this way when you finish a long rest. Wisdom is your spellcasting ability for all three spells.  In regards to the third and fourth questions, the fact that the PHB quote says &quot;use the spellcasting ability of that class when you cast the spell&quot; and the fact that the feats specify what ability the spell uses in these examples (independent of class) suggests to me that they are not associated with class. Any javascript event occurring when user clicks Stop load button? set of almost complex structures on $\mathbb R^4$ as two disjoint spheres The set of almost complex structures on $\mathbb R^{2n}$ is given by $$ M_n = \frac{GL(2n,\mathbb R)}{GL(n,\mathbb C)} = \mathcal C_+ \sqcup \mathcal C_-,$$ taking into account that $\det = \pm 1$ gives two disjoint sets.  How do we show that $M_2=S^2 \sqcup S^2$? Moreover, I'm a bit confused by dimensions: how does $\dim_{\mathbb R}M_2=8$ relates with the latter decomposition?  This is exercise 1.2.1 from the book Complex geometry by Huybrechts. Prove that $\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$ Using $\text{n}^{\text{th}}$ root of unity  $$\large\left(e^{\frac{2ki\pi}{n}}\right)^{n} = 1$$  Prove that   $$\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$$ What are the disadvantages of using a Zener diode over a linear voltage regulator? I wonder why a Zener diode is worse than linear voltage regulator.  Some of the posts online mentioned that because of the power dissipating problem. Zener diodes keep consuming a large amount of current while the load may just need a very little amount of current.   If I understand correctly, this phenomenon will happen only when the Zener diode is connected with the load in parallel. What if I connected my Zener diode with my load in series? Also, someone said that linear voltage regulators (e.g. LM7805) are not good voltage regulators as the power dissipated in them is much higher than that in switching voltage regulators. In my understanding, the power dissipated in a linear voltage regulator can be calculated by dropout voltage x current = power converted to heat.   Regarding these 2 points, other than a linear voltage regulator can handler a wider current range, what is the benefit of choosing a linear voltage regulator over a Zener diode? What is the list of keys and their corresponding wins?  I have it listing that the highest number of wins I've had in arena is a Champion key, but it doesn't say how many wins it was. fsck error on boot: /dev/sda6: UNEXPECTED INCONSISTENCY; RUN fsck MANUALLY I dual boot Ubuntu and Windows 7. I was browsing on FireFox in Ubuntu and all of a sudden I get some error message, so I rebooted.  Now when I try to boot into Ubuntu I get this message:  fsck from util-linux 2.26.2 /dev/sda6 contains a file system with errors, check forced. /dev/sda6: Inodes that were part of a corrupted orphan linked list found.  /dev/sda6: UNEXPECTED INCONSISTENCY; RUN fsck MANUALLY.         (i.e., without -a or -p options) fsck exited with status code 4 The root filesystem on /dev/sda6 requires a manual fsck  Busybox v1.22.1 (Ubuntu 1:1.22.0-15ubuntu1) built in shell (ash) Enter 'help' for a list of built-in commands.  (initramfs) _   How can I fix this? I have an iMac, connected to the Internet via Ethernet, and an iPhone 4. I would like to share the iMac's Internet connection to my iPhone via the USB Cable (because my wife dreams of a wifi-less household).  I've set up Internet Sharing, specifying "sharing from: Ethernet  via: USB IPhone", but when I test, the iPhone cannot connect to the Internet. Is there something I need to activate on the iPhone?   iMac, mac os x version: Mountain Lion, 10.8.4 iPhone 4, IOS Version: 6.1 How to revert to GNOME Classic Desktop? Validate/verify PDF files integrity Don't show pending suggested edits to me after I run out of votes What happens when I hover over a link in Chrome? Simplify $\arctan (\frac{1}{2}\tan (2A)) + \arctan (\cot (A)) + \arctan (\cot ^{3}(A)) $ How to simplify $$\arctan \left(\frac{1}{2}\tan (2A)\right) +  \arctan (\cot (A)) + \arctan (\cot ^{3}(A)) $$ for  $0&lt; A&lt; \pi /4$?  This is one of the problems in a book I'm using. It is actually an objective question , with 4 options given , so i just put $A=\pi /4$ (even though technically its disallowed as $0&lt; A&lt; \pi /4$) and got the answer as $\pi $ which was one of the options , so that must be the answer (and it is weirdly written in options as $4 \arctan (1) $ ).  Still , I'm not able to actually solve this problem. I know the formula for sum of three arctans , but it gets just too messy and looks hard to simplify and it is not obvious that the answer will be constant for all $0&lt; A&lt; \pi /4$. And I don't know of any other way to approach such problems. Partition around bad drive sectors? I've seen a lot of hard drives with only a few bad (unrecoverable) sectors (as indicated by SMART). Of course the proper thing to do with a bad drive is replace it, as there's a good chance of the drive crashing completely in the near future; I'm aware of that. However, in the interim, or for those who don't want to spend $70 to save data they don't value highly anyway (sigh, non-technical people), or just to get some use out of old hardware instead of throwing it away, I'm wondering if there's a way to identify where the unrecoverable sectors are ("where" meaning numerically, from 0 to the number of sectors on the drive), so I could stick a blank partition in that general area. Yes, this would involve low-level-formatting the drive and reinstalling the OS, of course.  So:  Would partitioning around unrecoverable sectors even be a workable way of reducing the disk of losing data?  If so, is there a way to find the location of these unrecoverable sectors? Subdivision surface modifier problem Hello!   I've recently started learning Blender and faced one problem.  After applying subdivision surface modifier I have plenty of extra faces. I've watched some teaching videos and I suppose additional faces shouldn't be added. Is it OK ? How to avoid this problem if it isn't. FileZilla: How to copy files from one directory to another on the same server? I want to copy files from one directory to another on the same server using FileZilla. Please guide me how can it be done? find a sequence converging to zero but not the element of lp space for every 1<_p<infinity What happens to the view.frame.size attribute after a CGAffineTransform (Rotate)?