  We find asymptotic equalities for the exact upper bounds of approximations by
Fourier sums of Weyl-Nagy classes $W^r_{\beta,p}, 1\le p\le\infty,$ for rapidly
growing exponents of smoothness $r$ $(r/n\rightarrow\infty)$ in the uniform
metric. We obtain similar estimates for approximations of the classes
$W^r_{\beta,1}$ in metrics of the spaces $L_p, 1\le p\le\infty$.
