Attached is a graph of a function of $x$ with three positive parameters $a_1$ $a_2$ $a_3$. The dashed horizontal line is $y=a_3$. Upon visual inspection of this graph, clearly this function approaches $y=a_3$ as x goes to infinity. However, looking at the function itself, I cannot see how this is happening:
$f(x;a_1,a_2,a_3)=(1+u)\left(\left(\frac{1+u}{u}\right)^{a_3}-1\right)$ where $u=\left(\frac{x}{a_1}\right)^{a_2}$
It seems to me that the limit as x goes to infinity should be zero. I'd appreciate it if someone can point out why the graph of the function is approaching $y=a_3$. (The parameter values in the graph are: $a_1=0.5$ $a_2=2.1$ and $a_3=1.5$)