In the meanwhile you figured it out, but nevertheless ...
In[130]:= Clear[fJT, liN]
fJT[x_] := 10 ArcTan[x/7]^3
liN[f_, x_, x0_?NumericQ] := Block[{fS},
If[FreeQ[f, x],
Print["The function ", f, " does not depend on ", x , ". Bye!"];
Return[$Failed], (* else *)
fS = D[f, x];
(f /. x -> x0) + (fS /. x -> x0) (x - x0)
]
]
In[133]:= liN[fJT[y], y, 7]
Out[133]= (5 \[Pi]^3)/32 + 15/112 \[Pi]^2 (-7 + y)
liN is the linearization searched for. Your super massive solve step
is mistaken, as you see reading the definition of L(x) again.
