  We present a new method that allows us to get a direct proof of the classical
Bernstein asymptotics for the error of the best uniform polynomial
approximation of $|x|^p$ on two symmetric intervals. Note, that in addition, we
get asymptotics for the polynomials themselves under a certain renormalization.
Also, we solve a problem on asymptotics of the best approximation of $\sgn(x)$
on $[-1,-a]\cup[a,1]$ by Laurent polynomials.
