find limits using L'Hopital rule and FTC

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, =\