  We derive the sharp Moser-Trudinger-Onofri inequalities on the standard
$n$-sphere and CR $(2n+1)$- sphere as the limit of the sharp fractional Sobolev
inequalities for all $n\ge 1$. On the $2$-sphere and $4$-sphere, this was
established recently by S.-Y. Chang and F. Wang. Our proof uses an alternative
and elementary argument.
