It should be:
Fernando Revilla
Please kindly check if there are any mistakes.
1. Find
Solution.
I use Fundamental Theorem of Calculus (FTC) to simplify the integral and then L'Hopital rule to find the limit:
2. Use L'Hopital's Rule to find the following limits:
(i)
(ii)
(iii)
Deduce the value of the limit .
Solution.
(i) substitute and take first derivative
(ii)
Take logarithm of the [tex]x^x{/math] and its limit:
and , see (i).
Therefore if , then the expression under log sign should be .
does this comment make sense?...
(iii)
Using L'Hopital Rule and FTC,
Here I am not sure at all whether I can take 0^0 - I rely on (ii) where I proved that .
Therefore, =\
It should be:
Fernando Revilla
Right.
Better, suppose:Therefore if , then the expression under log sign should be .
does this comment make sense?...
the, using the continuity of :
See my previous post. Besides, is an indetermined expression.
Fernando Revilla
I am glad it is! This was my first version (I think I took it from one of the books) then I started having doubts and added f(0) by analogy with f(b)-f(a) when the limits of integration are a and b. I guess I need to work it out myself to stop having doubts.
For , I googled for 0^0 and (1) google answered "1" and (2) there was this article which said "As a rule of thumb, one can say that 0^0 = 1".
sci.math FAQ: What is 0^0?
Conclusion: when it comes to mathematics, google is not a good source )))
Thanks as always for your reply!
When the expression "comes from trying to find a limit" you can't say . You need additional information about the corresponding functions.
Please, look at the proof of the Fundamental Theorem of Calculus.
Fernando Revilla
Just as a side note: there are way too many questions in the OP for one thread. See Rule # 8.