<s>[INST] Other methods for a limit I know that I can compute the limit
$$
\lim_{x\to1}\frac{Nx^{N+1}-(N+1)x^N+1}{(x-1)^2}=\frac{N(N+1)}{2}
$$
using L'Hospital's rule (not one but two times) but I am looking for other ways. Are there any of them?
p.s.: the limit follows from a shortcut used in order to find the value of $$
\sum_{k=1}^Nkx^k\Big|_{x=1}.
$$
 [/INST] You can simplify your fraction as
\begin{align}
\frac{Nx^{N+1}-Nx^N+1-x^N}{(1-x)^2} &=\frac{-Nx^N(1-x)+(1-x)(1+x+\dots+x^{N-1})}{(1-x)^2} \\
& = \frac{-Nx^N+1+x+\dots+x^{N-1}}{1-x} \\
& = \frac{1+x+\dots+x^{N-1}+x^N-(N+1)x^N}{1-x}
\end{align}
For $x \to 1$ the numerator tends to $0$, so this is still a form $0/0$ and you can apply de l'Hôpital:
\begin{align}
\lim_{x \to 1} -(1+2x+3x^2+\dots+Nx^{N-1}-N(N+1)x^{N-1})& =-\sum_{k=1}^Nk+N(N+1) \\
& = N(N+1)/2.
\end{align}
</s>