= Endpoint-constrained rational approximation to a power
{title2=$r(x)=x/[k+(1-k)x]$}
For $0<k\leq1$, impose $r(0)=0$, $r(1)=1$, and $r'(1)=k$ within the rational family $ax/(1+bx)$. Matching the first two Taylor coefficients at $x=1$ gives the displayed formula. It differs from the standard degree-$(1,1)$ <Padé approximant> to $x^k$ about one, which is $[(1-k)+(1+k)x]/[(1+k)+(1-k)x]$ and matches one further Taylor coefficient. At $k=0$, the constrained expression is interpreted by its removable constant limit.
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