Use the precise definition of a limit to show that $$\lim_{x\to 0} e^x=1.$$ By definition, I understand I need to show that: for every $\epsilon&gt;0$, there exists a $\delta&gt;0$ such that $0&lt;|x-0|&lt;\delta \implies |e^x-1|&lt;\epsilon$ for all $x$. I am able to show this, by finding such $\delta$, if I assume $(e^x-1)&gt;0$ but otherwise not. Any idea on how to proceed with the general case with the absolute value is appreciated.
We have been asked to prove the above statement through the two sub-proofs:   Firstly we must prove that $e^x$ is increasing and then prove that $\log(x)$ is also an increasing function.   After we are asked to prove that using the delta-epsilion definition of a limit that $\lim_{x \to 0} e^x = 1$.   I have done all of this but still don't see where the instructor is going with this.
Jackknife vs. LOOCV What are app domains used for? I'm on IBM AIX 7.1 where cURL and wget are not available. Also, I'm not in a position to install either.  Are there alternatives that would be available that would enable remote API calls using GET/POST? Copy current terminal prompt to clipboard Is there a way to copy the currently written content of the bash prompt? Say I wrote:  ping www.google.com   so that the lines in terminal looks like:  petr@sova:~$ ping www.google.com   and didn't hit return yet. I want to copy that text to clipboard. Is that possible?  The usual usecase would be hitting up several times and finding a command which I want to copy. List of Stored Procedure from Table I have a huge database with 100's of tables and stored procedures. Using SQL Server 2005, how can I get a list of stored procedures that are doing an insert or update operation on a given table. Is there to make the login screen appear on the external display in 18.04? Strictly Upper Triangular $n\times n$ Matrices form a Nilpotent Lie Algebra over $\mathbb{C}$ for $n\geq 2$ I have been attempting this problem for a while, it is an assignment problem so I don't want somebody to just post the answer, I'm just looking for hints.  Let $\mathfrak{u}(n,\mathbb{C})$ be the Lie algebra of strictly upper triangular $n\times n$ matrices over $\mathbb{C}$.  I am trying to show that $\mathfrak{u}(n,\mathbb{C})$ is a nilpotent Lie algebra for $n\geq 2$.  This is what I have looked at so far.  I have done some explicit calculations (I shall omit the bulk of them and just summarise below) for $n=2,3,4$ and have come up with some ideas.  Let $L=\mathfrak{u}(n, \mathbb{C})$.  If $n=2$,  $L^1 = [L, L] = &lt;0&gt;_{\mathbb{C}}$  So $L$ is nilpotent.  If $n=3$,  $L^1 = [L, L] = &lt;e_{13}&gt;_{\mathbb{C}}$  $L^2 = [L, L^1] = &lt;0&gt;_{\mathbb{C}} = \left\{ 0 \right\}$  So, $L$ is nilpotent.  If $n=3$  $L^1 = [L,L] = &lt;e_{13},e_{14},e_{24}&gt;_\mathbb{C}$  $L^2 = [L,L^1] = &lt;e_{14}&gt;_{\mathbb{C}}$  $L^3 = [L, L^2] = &lt;0&gt;_{\mathbb{C}} = \left\{ 0 \right\}$  So, $L$ is nilpotent.  So my hypothesis so far is that for an arbitrary $m\in\mathbb{N}$, $L^{m-1}=\{0\}$  My instinct is to do this by induction. My base case, $n=2$ has already been shown. Now I assume that $\mathfrak{u}(n,\mathbb{C})$ is nilpotent for $n=k$ ($k\in\mathbb{N}$) and consider the case for n=k+1.  However, I have no idea how to link these two cases together. I have read in Introduction to Lie Algebras and Representation Theory - J.E. Humphreys that $L^k$ should be spanned by basis vectors $e_{ij}$ where $j-i=k+1$ - however in my calculations for $L^1$, I obtain an $e_{14}$ which does not satisfy 4-1=2.  Am I missing something completely obvious? Or have I made a mistake somewhere? (I have checked my calculations numerous times, but cannot find any errors!)  Thanks,  Andy. How to insert a figure between two texts? I'd like to create something similar to what is give below: text on either side of a vertically centred image: After reading an article about Ubuntu's netflix support I decided to try running netflix through chromium but I got this error code: M7063-1913 I assumed that it was a problem with html5 but in the playback preferences it doesn't give me the option to use it. Are Mersenne prime exponents always odd? I have been researching  so I can write a program that finds them. A Mersenne prime looks like $2^n-1$. When calculating them, I have noticed that the $n$ value always appears to be odd. Is there confirmation or proof of this being true? Replacements for switch statement in Python? Is it a wise-decision to buy full-frame lenses for use on an APS-C body? I have a Canon 550D which I've been using for around a year now. I really love it and have learned a great deal with this and the 18 - 135mm kit lens. But now with experience I feel the need to get some better glass. I won't be buying a new body any time soon because I figure investing in some good glass is more important.   As I have looked around, I see quite a few good used L lenses I can get in ebay for good bargain prices. But I know that the EF lenses don't have the same focal length in the cropped sensors. Is it a wise decision to buy these EF lenses or buy the cropped sensor alternatives?  I have the following lenses in mind that I am looking to buy down the line from the canon range over the next several months.  16-35mm f/2.8L, 24-70mm f/2.8L, 70-200 f/4L.  Are these apt choices for cropped sensors? I know most pros with FFs use these but I am a bit skeptic that they will serve the same purpose for me. If so what are the alternatives for this that have almost as good image quality? Prove that some topology is not metrizable Counter example for $(A \times B) \cap (C \times D) = (A \cap C ) \times (B \cap D)$ Exclusive differences Profiles vs permission sets? I understand, the differences between Profiles and Permission Sets. I just wanna ask are there some configs that make Profile or Permission Set unique.  Are there things that could only be done via permission set and not via Profiles and vice versa ? How to show $\sum_{i=1}^{n} \binom{i}{2}=\binom{n+1}{3}$? Show that $\,\displaystyle\sum_{i=1}^{n} \binom{i}{2}=\binom{n+1}{3}$.   I'm thinking right now (though not getting anywhere with it) that I want to expand out the summation portion to $i!/2!(i-2)!$ and simplify from there? Not sure if that will help, not to mention if I put $1$ in for $i$ I get $1/-2$ which I don't think is right.   Anyone care to shed some light on the subject?   Thanks. I'm attempting to modify the Table of Contents in my document. Currently, chapter one in my document is listed as   1   Introduction   What I would like to see is  Chapter 1   Introduction   Seems like an easy enough thing to ask for, but can't seem to find a good way to do this. Any feedback would be most appreciative. How do I restrict my kids' computing time? Connecting to sql server database mdf file without installing sql server on client machine? Pie Chart without labels in beamer using pgf-pie