some questions about a solution (integral)

[Vali]

\(\displaystyle \lim_{n \to \infty }\int_{0}^{\frac{\pi}{3}}\frac{sin^{n}(x)}{sin^{n}(x)+cos^{n}(x)}dx\)
I found a solution on the internet but I don't understand it completely.

I understood the first part ( they switch the limit with the integral ).I just don't understand the values of the limits.
Also, I understood those conditions and intervals.The 1/2 limit I think I understood it.If sinx=cosx the limit is like I would have (n/2n) as n->00 so the limit is 1/2, right?
Also, the last limit which is 0.For example if I take x=pi/6.At numerator I would have (1/2)^n which is 0 because -1<1/2<1 so the whole limit is 0.That's the reason why the last limit is 0?
But why the first limit is 1?

 

[Steven G]

Why like n/(2n)? How about exact. sinⁿ(x)/(2sinⁿ(x)) = 1/2

Also, the last limit which is 0.For example if I take x=pi/6.At numerator I would have (1/2)^n which is 0 because -1<1/2<1 so the whole limit is 0.That's the reason why the last limit is 0? What happens if the denominator also goes to 0?? Did you check for that? After all 0/0 \(\displaystyle \neq\) 0

I would graph sin(x) and cos(x) from 0 to pi/3 and look at the two graphs closely where sin(x)>cos(x) and sin(x)<cos(x)

 

[Vali]

I understood.
For x from (pi/4,pi/3) sinx>cosx so tgx>1 so limit of 1/tg^n(x)=0.So if /sin^n(x)) the initial function becomes 1/(1+1/tg^n(x)) so the limit is 1.
For x from (0,pi/4) sinx<cosx so limit of tg^n(x)=0.If /:cos^n(x) the initial function becomes tg^n(x)/[1+tg^n(x)] so the limit is 0, right?