  Using Chebyshev polynomials combined with some mild combinatorics, we provide
a new formula for the analytical planar limit of a random matrix model with a
one-cut potential $V$. For potentials $V(x)=x^{2}/2-\sum_{n\ge1}a_{n}x^{n}/n$,
as a power series in all $a_{n}$, the formal Taylor expansion of the analytic
planar limit is exactly the formal planar limit. In the case $V$ is analytic in
infinitely many variables $\{a_{n}\}_{n\ge1}$ (on the appropriate spaces), the
planar limit is also an analytic function in infinitely many variables and we
give quantitative versions of where this is defined.
  Particularly useful in enumerative combinatorics are the gradings of $V$,
$V_{t}(x)=x^{2}/2-\sum_{n\ge1}a_{n}t^{n/2}x^{n}/n$ and
$V_{t}(x)=x^{2}/2-\sum_{n\ge3}a_{n}t^{n/2 -1}x^{n}/n$. The associated planar
limits $F(t)$ as functions of $t$ count planar diagram sorted by the number of
edges respectively faces. We point out a method of computing the asymptotic of
the coefficients of $F(t)$ using the combination of the \emph{wzb} method and
the resolution of singularies. This is illustrated in several computations
revolving around the important extreme potential
$V_{t}(x)=x^{2}/2+\log(1-\sqrt{t}x)$ and its variants. This particular example
gives a quantitive and sharp answer to a conjecture of t'Hoofts which states
that if the potential is analytic, the planar limit is also analytic.
