Let's start from the limit...
(1)
First we construct the Taylor expansions...
(2)
(3)
... and then we devide (2) and (3) obtaining the result...
(4)
Marry Christmas from Serbia
I need to solve them without Lhopital's rule and Taylor expansion, becouse one of my friends has these problems during first semester of studying maths, and they don't know these technics yet. I solved couple of similar problems using the fact,
that if you have then you can substitute with when
and
and then calculate
. I can find these functions for
and but this doesn't help if i don't know these functions for
and
An alternative approach [I do hope it is 'allowed'...] to the second integral uses the 'infinite product'...
(1)
Setting in (1) and with a little computation we obtain...
(2)
... and from (2), taking into account the 'fundamental limit'...
(3)
... we obtain [finally!]...
(4)
Marry Christmas from Serbia