The classical Hardy inequality holds in Sobolev spaces $W_0^{1,p}$ when $1 \le p < N$. In the limiting case where $p=N$, it is known that by adding a logarithmic function to the Hardy potential, some Hardy type inequality which is called the critical Hardy inequality holds in $W_0^{1,N}$. In this talk, in order to find a reason why the logarithmic function appears at the potential, we derive the logarithmic function from the form of the classical Hardy inequalities with the best constants by a suitable limiting procedure as $p \nearrow N$ via extrapolation. And we also discuss the second order case which is called the Rellich inequality. This is a joint work with Prof. Sobukawa (Waseda University).