For Problem 3 of Section 2.1, we get an asymptotic expression for how a specific solutions evolves in t at a fixed x.  For comparison, we plot the exact solution and the asymptotic expression together.

w[x_,t_]=(Erf[x/Sqrt[4 t]]+1)/2

Plot[w[10,t],{t,0,1000},PlotRange->{0,1}]

asy[x_,t_]=(1+2 x/Sqrt[4Pi t])/2

Plot[{asy[10,t],w[10,t]},{t,0,1000},PlotRange->{0,2}]

Note that the asymptotic expression matches the exact solution as t->∞.  An advantage of the asymptotic expression is that it can be evaluated using only simple operations.  As usual with approximations, this represents a trade-off of exactness for simplicity.