The attached figure is a graph of a function of $x$ with three positive parameters $a_1$, $a_2$, and $a_3$. 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 $a_3=1.5$)
You can write $\left(\frac{1+u}u\right)^{a_3}=\left(1+\frac 1u\right)^{a_3}\approx 1+\frac {a_3}u$, where the approximation is valid as $u$ (and therefore $x) \to \infty$. Then your function goes to $a_3$ as your graph does.
Added: Inside the outer parentheses you then have $1+\frac {a_3}u -1=\frac{a_3}u$. This is multiplied by the outer $(1+u)$, giving $a_3+\frac {a_3}u$. The first term is your constant.