  Scalar field theories with quartic interaction are quantized on fuzzy $S^2$
and fuzzy $S^2\times S^2$ to obtain the 2- and 4-point correlation functions at
one-loop. Different continuum limits of these noncommutative matrix spheres are
then taken to recover the quantum noncommutative field theories on the
noncommutative planes ${\mathbb R}^2$ and ${\mathbb R}^4$ respectively. The
canonical limit of large stereographic projection leads to the usual theory on
the noncommutative plane with the well-known singular UV-IR mixing. A new
planar limit of the fuzzy sphere is defined in which the noncommutativity
parameter ${\theta}$, beside acting as a short distance cut-off, acts also as a
conventional cut-off ${\Lambda}=\frac{2}{\theta}$ in the momentum space. This
noncommutative theory is characterized by absence of UV-IR mixing. The new
scaling is implemented through the use of an intermediate scale that demarcates
the boundary between commutative and noncommutative regimes of the scalar
theory. We also comment on the continuum limit of the $4-$point function.
