Let $(u_n)_{n\in\mathbb{N}}$ with the general term $u_n=\frac {1+x^{n}}{1+x+x^{2}+...+x^{n+p-1}}$, where $x\ge0$ and $p \in \mathbb{N}$. Let $f(x)= \lim_{n\to\infty}u_n$. Find the differentiability and continuity domain.
First I tried to simplify a little $u_n$ using the sum of the geometric progression and I got this $$u_n=\frac{(1+x^{n})(x-1)}{x^{n+p}-1}.$$ So if $x \in (0,1)$, then $f(x)=1-x$.
What should I do when $x = 1$ and $x>1$?
First of all, I think you have a slight error in you the sum of a geometric progression formula: the denominator must be $x^{n+p}-1$, not $x^{n+p\color{red}{-1}}-1$.
If $x=1$, you can literally plug it into the original expression for $u_n$.
If $x>1$, then the let's divide the numerator and denominator by $x^n$: $$u_n=\frac{(1+x^n)(x-1)}{x^{n+p}-1}=\frac{\left(\frac{1}{x^n}+1\right)(x-1)}{x^p-\frac{1}{x^n}},$$ and observe that in this case $1/x^n\to0$ as $n\to\infty$.