I have the following recurrence condition for $t \geq 2$
$$x_t = c_1\left(-\frac{1}{x_{t-1}} + \sqrt{\frac{1}{x_{t-1}^2} + c_2}\right)$$
Let $x_1 = c_3$. Now, i would like to obtain the growth of the sequence $x_t$ or in simple terms whats the convergence rate , also $c_1,c_2,c_3 > 0$ are constants.
The convergence rate is defined as
$$ \lim_{t\rightarrow\infty}\frac{\left|x_t-L\right|}{\left|x_{t-1}-L\right|} $$
If the sequence converges to $0$, $L=0$. I do not believe this is the case in general, but for some combination of constants, this is true. Maybe this will lead you to a way of a general proof. Notice that
$$ x_t=c_1\left(-\frac{1}{x_{t-1}}+\sqrt{\frac{1}{x_{t-1}^2}+c_2}\right)=\frac{c_1c_2}{\sqrt{\frac{1}{x_{t-1}^2}+c_2}+\frac{1}{x_{t-1}}} $$
So, divide by $x_{t-1}$ to get
$$ \frac{x_t}{x_{t-1}}=\frac{c_1c_2}{\sqrt{1+c_2x_{t-1}^2}+1} $$
The limit as $t\rightarrow\infty$ is $\frac{c_1c_2}{2}$. At the very least, we know that the convergence rate cannot exceed $\frac{c_1c_2}{2}$