@ flyingsquirrel : great
Here is an alternative method :
The limit is now
Thus the limit is...
@ Mathstud, #32 : good ^^
Now by definition of the derivative
And we end this marathon.
EDIT: Wow, I sounded really really arrogant in the beginning. Sorry everyone unintentional I assure you.
I like seeing others do these, but I thought I would throw my hat in.
I will take a stab at one. But I like to watch mathstud, Moo, and others go at them. Then I can absorb some of their 'useful tricks'.
Here is a method I used for this particular integral some years back. I have seen it before.
It just entails a little algebra and e.
Here is #4 using a little algebra and a substitution.
Then, we note that as ,
Make the sub and we get:
Now, a little algebra and 'e'; add 1 and subtract 1:
Because of asymptotics that +1 is just a 'hanger on' and we can rewrite as such using e:
Now, the limit, which is the exponent of e, can be taken:
Now, the limits here I won't bother deriving. I will use it as a 'lemma'
Analogous with b, which is ln(b).
So, we end up with:
Which leads to