I used substitution $$t=x-1, x=t+1,x\rightarrow1\Rightarrow t\rightarrow 0$$  Now the expression is $$\lim_{t\to\ 0}\frac{t+1+(t+1)^2+\cdots+(t+1)^n-n}{t}$$  Can we use the sum of geometric sequence or squeeze theorem?
I am able to evaluate the limit $$\lim_{x \to 1} \frac{x + x^2 + \dots + x^n - n}{x - 1} = \frac{n(n + 1)}{2}$$ for a given $n$ using l'Hôspital's (Bernoulli's) rule.  The problem is I don't quite like the solution, as it depends on such a heavy weaponry. A limit this simple, should easily be evaluable using some clever idea. Here is a list of what I tried:   Substitute $y = x - 1$. This leads nowhere, I think. Find the Taylor polynomial. Makes no sense, it is a polynomial. Divide by major term. Dividing by $x$ got me nowhere. Find the value $f(x)$ at $x = 1$ directly. I cannot as the function is not defined at $x = 1$. Simplify the expression. I do not see how I could. Using l'Hôspital's (Bernoulli's) rule. Works, but I do not quite like it.   If somebody sees a simple way, please do let me know.    Added later: The approach proposed by Sami Ben Romdhane is universal as asmeurer pointed out. Examples of another limits that can be easily solved this way:   $\lim_{x \to 0} \frac{\sqrt[m]{1 + ax} - \sqrt[n]{1 + bx}}{x}$ where $m, n \in \mathbb{N}$ and $a, b \in \mathbb{R}$ are given, or $\lim_{x \to 0} \frac{\arctan(1 + x) - \arctan(1 - x)}{x}$.   It sems that all limits in the form $\lim_{x \to a} \frac{f(x)}{x - a}$ where $a \in \mathbb{R}$, $f(a) = 0$ and for which $\exists f'(a)$, can be evaluated this way, which is as fast as finding $f'$ and calculating $f'(a)$.  This adds a very useful tool into my calculus toolbox: Some limits can be evaluated easily using derivatives if one looks for $f(a) = 0$, without the l'Hôspital's rule. I have not seen this in widespread use; I propose we call this Sami's rule :).
Pages: How to put a word between horizontal rules How can I make something that looks like this?  --------   Text   ---------  where the horizontal rules extend to the margin on either side? Get Label value for specified language Delete all files except in a certain subdirectory with find I want to recursively delete all files not accessed in a while in folder a, except all files in the subfolder b.  find a \( -name b -prune \) -o -type f -delete   However, I get an error message:     find: The -delete action automatically turns on -depth, but -prune does   nothing when -depth is in effect.  If you want to carry on anyway,   just explicitly use the -depth option.   Adding -depth causes all files in b to be included, which must not happen.  Anyone know a safe way to make this work? If $\int_0^1 f(x)x^n \ dx=0$ for every $n$, then $f=0$. This problem has been seen on mid-2012 MacBook Pros on Mavericks and on late-2014 Retina MacBook Pros on Yosemite.  One day while using iMessage on the Mac, all the contacts turned red and messages failed to send, immediately returning a "not delivered" error. It also reported when trying to resend the messages that the contacts were no longer associated with an iMessage account.  Things that didn't fix this behavior:   Deleting the com.apple.ids.madrid Removing the Apple ID information from my phone, turning off iMessage on both of the devices, and placing my iPhone in airplane mode for several minutes before turning it on logging in again.   Could someone help me figure out how to solve this? Mysterious! Beamer itemize has no bullets after adding a few packages this is my first time using beamer.  I copied a template from the sharelatex page on Beamer  Which works, and \begin{itemize} shows bullets     But after adding a few more packages that I need, all the bullets are gone!    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