I have a function of geometric series \begin{equation} f(\lambda,x) = \frac{1-\lambda^{x+1}}{1-\lambda^x}, \end{equation} where $|\lambda|<1$ and $x\in Z^+$.
For a fixed $x$, i.e. $x=500$, $f(\lambda,x)$ will start from 1 and converge to 1.002, as $\lambda$ moves towards $1$.
I am wondering, is it possible to find a closed-form solution as a function of $x$ to determine $\lim_{\lambda\rightarrow 1} f(\lambda,x)$?
P.S. it seems that $\lim_{\lambda\rightarrow 1} f(\lambda,x) = 1 + \frac{1}{x}$, but why?