An interesting exercise to show the condition of a function:

#14, p. 30: assume a relative error in x of xhat=x(1+h). Looking at the  relative error in the Log function, we can simplify to get

Clear[x,h,f,g,re]

We can compute the relative error directly: Mathematica doesn’t always simplify as well as we can!

FullSimplify[Log[1+x*(1+h)]-Log[1+x]]

f[x_]:=Log[1+x*h/(1+x)]
g[x_]:=Log[1+x]
re[x_]:=f[x]/g[x]

Limit[Simplify[re[x]/h],h->0]

Condition gives the same thing (with Abs):

condition[x_]:=Abs[x*g'[x]/g[x]]
condition[x]

Plot[condition[x],{x,-1,1}]