Why the following is not correct? : $$\lim_{n \to \infty} (1+\frac1n)^n = \lim_{n \to \infty} (1+\frac1n) \lim_{n \to \infty} (1+\frac1n) \dots \lim_{n \to \infty} (1+\frac1n) = 1 \times 1 \times \dots 1 = 1.$$ I guess that the problem (as usual) lies in infinity, but if so how? I checked all the theorems in Fitzpatrick's book of Calculus, nothing prevents to do the limits infinitely many times (esp. for a convergent one)!
On Wikipedia, it says that $\lim_{n\to\infty}(1+\frac{1}{n})^n = e$ :     It [e] is approximately equal to 2.71828,[1] and is the limit of (1 + 1/n)n as n approaches infinity, ... ()   When I evaluate $(1+\frac{1}{n})^n$ for $n = 10^8$, I get approximately $2.718281798347$ which indeed is pretty close to $e$.  But when I try to "solve" the limit using the laws of limits, I get  $$\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n = \left(\lim_{n\to\infty}\left(1+\frac{1}{n}\right)\right)^n$$  because of the power law  $$ = \left(\lim_{n\to\infty}(1)+\lim_{n\to\infty}\left(\frac{1}{n}\right)\right)^n$$  because of the addition law  $$=\left(1+0\right)^n = 1$$  but that would mean that $e=1$, which is obviously not true.  What am I missing / doing wrong?  Thanks in advance.
Show all inbox items (notifications) in app The SE app currently only shows some inbox items:   Answers Comments Chats   When I see those I assume I'm all caught up, but it turns out there are more on the web which aren't displayed in the app:   Bounty expiration Elections Edits on your own posts Maybe others?   I was about to tag this  but realized it's probably intentional.  I imagine the reason for not showing them is because they're not something you can act on from within the app. Bounties can't be offered or awarded1 and elections can't be voted on.  But then again chats can't be replied to either. Badge descriptions can't be viewed (on the Achievements side). Selecting those launches the browser. Why then shouldn't bounties or elections launch the browser too? Hiding them just means I'll miss it if I haven't visited in browser recently.  1 Though accepting answers does award the bounty JSON+Node.js - Unexpected token o Vector. Properties It is given in a question that $\vec x+\vec a=(\vec a\cdot \vec x)\vec b$ and it is stated that $\vec a$, $\vec b$ and $\vec x$ are vectors.  It is also stated that $\vec a\cdot \vec b\ne 1$.  The question asks to express $\vec x$ in term $\vec a$ and $\vec b$.  How do you this? My public website name and AD domain name are the same. How can I get to my external website from inside my network? What is the difference between therapy and treatment? how many conditional sentence types are possible? I wonder how many conditional sentence types are possible. Basically, there are four main and two mixed types. By mixing tenses, it could be up to nine of them. Are the rest viable?  I came across the following examples in Full Spate Pre-Profficiency Primer:   After you have just sold alcohol to a young guy. If he is if under eighteen, you should not have sold him the alcohol. If he promised to bring the drinks, he will. If you will take a seat, the doctor will see you shortly. (Likely, the former will demonstrates politeness only.) I would be grateful if you would tell me a little about your friends. (The same politeness, but would instead.) If he will keep eating all those biscuits, he can not expect to stay as slim as he used to be.   I would be grateful if someone could shed more light on the topic. I'm getting these errors ramdomly, and I don't know if its normal or not.   [39441.061856] ata3.00: failed to read SCR 1 (Emask=0x40) [39441.061866] ata3.01: failed to read SCR 1 (Emask=0x40) [39441.061892] ata3.15: exception Emask 0x10 SAct 0x0 SErr 0x280100 action 0x6 frozen [39441.061897] ata3.15: irq_stat 0x08000000, interface fatal error [39441.061904] ata3.15: SError: { UnrecovData 10B8B BadCRC } [39441.061910] ata3.00: exception Emask 0x100 SAct 0x0 SErr 0x0 action 0x6 frozen [39441.061917] ata3.01: exception Emask 0x100 SAct 0xe SErr 0x0 action 0x6 frozen [39441.061923] ata3.01: failed command: READ FPDMA QUEUED [39441.061933] ata3.01: cmd 60/a8:08:b0:48:62/00:00:00:00:00/40 tag 1 ncq 86016 in [39441.061940] ata3.01: status: { DRDY } [39441.061944] ata3.01: failed command: READ FPDMA QUEUED [39441.061953] ata3.01: cmd 60/a8:10:b0:49:62/00:00:00:00:00/40 tag 2 ncq 86016 in [39441.061959] ata3.01: status: { DRDY } [39441.061963] ata3.01: failed command: READ FPDMA QUEUED [39441.061972] ata3.01: cmd 60/58:18:58:4a:62/00:00:00:00:00/40 tag 3 ncq 45056 in [39441.061978] ata3.01: status: { DRDY } [39441.061987] ata3.15: hard resetting link [39441.608302] ata3.15: SATA link up 3.0 Gbps (SStatus 123 SControl 300) [39441.609090] ata3.00: hard resetting link [39441.929246] ata3.00: SATA link up 3.0 Gbps (SStatus 123 SControl 320) [39441.929333] ata3.01: hard resetting link [39442.249184] ata3.01: SATA link up 3.0 Gbps (SStatus 123 SControl 320) [39442.263242] ata3.00: configured for UDMA/133 [39442.277570] ata3.01: configured for UDMA/133 [39442.277725] ata3: EH complete   I'm also pasting smartctl -a for ,  and .  Thanks in advance for your help. How to make 7-zip do a whole bunch of folders Unable to lock the administration directory (/var/lib/dpkg/) is another process using it? I get this error when trying to use apt-get:  E: Could not get lock /var/lib/dpkg/lock - open (11 Resource temporarily unavailable) E: Unable to lock the administration directory (/var/lib/dpkg/) is another process using it?     How can I fix this? If the plural of ‘man’ is ‘men,’ shouldn’t the plural of ‘German’ be ‘Germen’? What makes these two words so different that 'man' is changed to 'men', but 'German' is changed to 'Germans'? Is my phone liable to explode if I charge it with a charger that came with a different phone? My friend and I both have droids and we often user each other's wall chargers out of convenience. I was told that it's possible that my phone might explode if it's charged with a charger that wasn't built specifically for my phone. How true is this across droids? Why does the browser renders a newline as space? I just bought a controller because I am tired of not to being able to do any tricks with keyboard controls. The problem is I am can't configure controls as FIFA 13 PC version  doesn't consider a controller in their settings.  Does anyone know how to do this? Alien studying human race for mental capacity and ESP Where can I get the original Unix? This button is in the new Firefox, but what is the name of it? What is 'Currying'? I've seen references to curried functions in several articles and blogs but I can't find a good explanation (or at least one that makes sense!) Given ∃x.¬p(x), use the Fitch System to prove ¬∀x.p(x). Why does the Community user approve and reject edits? Why does the Community user sometimes approve or reject edits? How does a machine know if my edit is valid? What does it mean if my edit was rejected by Community?     See also: What happens to a polymorphed character if he is shorn as a sheep?