How can I evaluate the following limit? $$\lim_{t\to \infty} \Gamma(t+1)-\frac{\Gamma'(t+1)L(t)}{\sqrt{1+\Gamma'(t+1)^2}}$$ Where $$L(t)=\int_0^t \sqrt{\Gamma'(x+1)^2+1}dx$$ ?
The reason I want to know is this: I was playing around with the involutes of curves and I decided to graph the involute of $y=\Gamma(x+1)$. The involute looks like this:
As you can see, the $y$ value seems to converge and be asymptotic. And so the limit that I am trying to evaluate is the limit of the $y$ coordinate of the parametric equation as $t \to \infty$. Just thought I'd add a little bit of context.
Does anybody have any idea how to evaluate such a limit? Or even to show that it converges to a value? I have no idea how to even start with it.
Any help is much appreciated!