  We present results for SCET and bHQET matching coefficients and jet functions
in the large-$\beta_0$ limit. Our computations exactly predict all terms of the
form $\alpha_s^{n+1} n_f^n$ for any $n\geq 0$, and we find full agreement with
the coefficients computed in the full theory up to $\mathcal{O}(\alpha_s^4)$.
We obtain all-order closed expressions for the cusp and non-cusp anomalous
dimensions (which turn out to be unambiguous) as well as matrix elements (with
ambiguities) in this limit, which can be easily expanded to arbitrarily high
powers of $\alpha_s$ using recursive algorithms to obtain the corresponding
fixed-order coefficients. Examining the poles laying on the positive real axis
of the Borel-transform variable $u$ we quantify the perturbative convergence of
a series and estimate the size of non-perturbative corrections. We find a so
far unknown $u=1/2$ renormalon in the bHQET hard factor $H_m$ that affects the
normalization of the peak differential cross section for boosted top quark pair
production. For ambiguous series the so-called Borel sum is defined with the
principal value prescription. Furthermore, one can assign an ambiguity based on
the arbitrariness of avoiding the poles by contour deformation into the
positive or negative imaginary half-plane. Finally, we compute the relation
between the pole mass and four low-scale short distance masses in the
large-$\beta_0$ approximation (MSR, RS and two versions of the jet mass), work
out their $\mu$- and $R$-evolution in this limit, and study how their
implementation improves the convergence of the position-space bHQET jet
function, whose three-loop coefficient in full QCD is numerically estimated.
