Q: a limit of recurrence relation I have and the following recurrence relaation:
, 
I need to solve 
I found that this string is increasing and I tried to find z and w from general form 
I tried to factorize xn but I didn't get too far.

A: You can prove by induction that $x_n\geq 1$. This implies that for all $n\geq 0$, $$x_{n+1}-x_n\geq \frac 1 a$$
Summing this up leads to $$x_n\geq 1 + \frac n a$$
So, clearly, $x_n\rightarrow +\infty$ as $n\rightarrow +\infty$.
The trick is to look for an estimate of $u_n = x_n^a$. 
Note that just like $(x_n)_{n\geq 0}$, $(u_n)_{n \geq 0}$ tends to $+\infty$, so we can use the appropriate Taylor expansion:
$$\begin{split}
u_{n+1} &= \left (x_n + \frac 1 a x_n^{1-a}\right)^a\\
&= u_n \left (1 + \frac 1 a \frac 1 {u_n}\right)^{a}\\
&= u_n \left (1 + \frac 1 {u_n} + o\left (\frac 1 {u_n}\right)\right)\\
&= u_n +1 +o(1)
\end{split}$$
So summing this up leads to $u_n=n+o(n)$, which implies that
$$\lim_{n\rightarrow +\infty} \frac {x_n^a}n = 1$$
