Limits of integrals How would you show that if
$f : [0,1] \rightarrow \Bbb R$ is continuous, then
$$\lim_{n\rightarrow \infty}\int_0^1\int_0^1 \cdots \int_0^1 f\left( \frac{x_1+x_2+\cdots+x_n}{n} \right)~dx_1~dx_2\cdots dx_n = f\left( \frac{1}{2} \right) $$
and
$$\lim_{n\rightarrow \infty}\int_0^1\int_0^1 \cdots \int_0^1 f((x_1 x_2 \cdots x_n)^{\frac{1}{n}})~dx_1~dx_2\cdots dx_n = f\left(\frac{1}{e}\right) $$

For the second problem, recall that Weierstrass tells us that polynomials are dense in $C[0,1].$ So it's enough to  prove the result for polynomials, and for this it's enough to prove it for each $x^k.$ For $x^k$ the $n$-fold integral works out nicely to be $1/(1+k/n)^n.$ This has limit $1/e^k = (1/e)^k,$ which is the desired answer for this function.
Weiersrtass can also be used for the first problem as well, although it's not quite as simple.
