The transient temperature field resulting from a constant and uniform temperature
T
s
{T_s}
(or time-dependent heat flux
H
=
h
t
−
1
/
2
H = h{t^{ - 1/2}}
) imposed at the surface of a halfspace initially at uniform temperature
T
0
{T_0}
is considered. A temperature-dependent thermal conductivity variation,
k
(
T
)
=
k
0
exp
[
λ
(
T
−
T
0
)
/
T
0
]
k\left ( T \right ) = {k_0}\exp \left [ {\lambda (T - {T_0})/{T_0}} \right ]
, and a constant product of density and specific heat,
ρ
C
\rho C
, are assumed to be accurate models for the halfspace for some useful temperature range. The problem is initially formulated in terms of the dimensionless conductivity
ϕ
=
k
(
T
)
/
k
0
\phi = k\left ( T \right )/{k_0}
. Attention is then focused on the singular problem resulting from the limits
ϕ
s
=
ϕ
(
T
s
)
↓
0
{\phi _s} = \phi \left ( {{T_s}} \right ) \downarrow 0
and
ϕ
s
→
∞
{\phi _s} \to \infty
. This work considers the use of matched asymptotic expansions to solve the problem under the first of these limits. In particular, Fraenkel’s interpretation [5] of Van Dyke’s method of inner and outer expansions [6] is carefully applied to the problem under consideration. Besides obtaining a uniformly valid solution to the problem, a particularly interesting explicit result is deduced, namely
\[
lim
ϕ
s
↓
0
h
=
−
(
1.182754
⋅
⋅
⋅
)
(
T
0
/
λ
)
[
ρ
C
k
0
/
2
]
1
/
2
+
O
(
ϕ
s
l
n
ϕ
s
)
\lim \limits _{{\phi _s} \downarrow 0} h = - (1.182754 \cdot \cdot \cdot )({T_0}/\lambda ){[\rho C{k_0}/2]^{1/2}} + O({\phi _s}ln{\phi _s})
\]