Hey guys, Look at this limit

If I use L'Hopital's rule directly, I can get the limit is 1/2

but as I tried to solve the problem in another approach, the question occured,

let and

then the limit could be written as

and since

the term could then be written as

and now use the L'Hopital' rule,

we will get the limit is 1

I know there is something wrong, but would you mind pointing out where I have made mistakes? thanks.