Find :$ \lim_{n\rightarrow \infty} \int_{0}^{1} f(x^n) dx $? 
Given  $f$  is  continuous  on $[0,1]$ , Then $$ \lim_{n\rightarrow \infty} \int_{0}^{1} f(x^n) dx  $$ is 
  
  
*
  
*$ f(1)$
  
*$f(0)$
  
*$1$
  
*$0$

I thinks  it will be  $0$ because    $ \lim_{n\rightarrow \infty} \int_{0}^{1} x^n dx = \lim_{n\rightarrow \infty}\frac{1}{n+1} = 0  $
Is its true ?

The limit is $f(0)$ by DCT. A dominating integrable function is the supremum of $|f|$. 
