I'm trying to do a problem but I can't figure out how to do it. We have to calculate a limit using Landau notation. I'm in the french schooling system so I'm not sure if this is what it is called but I mean domination, preponderance and equivalence of functions around a certain value writen with the symbols O , o and ~ respectively. I also know some basics of how to use Taylor Young developpements but was explicitly told not to use Hopital's rule.
Anyway, with all of that out of the way, here is the limit:
$$\lim_{t \to 1}{(\frac{a+t}{a+1})^{\frac{t}{1-t}}} $$
with a as constant Thank you for your time
Using that $\log(1 + x) = x + o(x)$ when $x\to 0$ $$ \log\lim_{t\to 1}{\left(\frac{a+t}{a+1}\right)^{\frac{t}{1-t}}} = \lim_{t\to 1}\log{\left(\frac{a+t}{a+1}\right)^{\frac{t}{1-t}}} =$$ $$ \lim_{t\to 1}\frac{t}{1-t}\,\log{\left(\frac{a+t}{a+1}\right)} = \lim_{t\to 1}\frac{t}{1-t}\,\log{\left(\frac{(a+1) + (t-1)}{a+1}\right)} = $$ $$ \lim_{t\to 1}\frac{t}{1-t}\,\log{\left(1 + \frac{t-1}{a+1}\right)} = \lim_{t\to 1}\frac{t}{1-t}\,\left(\frac{t-1}{a+1} + o(t-1)\right) =\cdots $$