ca.analysis and odes - On the extension of a limit

We know that $lim_{prightarrowinfty}leftVert left(x_{1},cdots,x_{n}right)rightVert _{p}=maxleft{ left|x_{1}right|,cdots,left|x_{n}right|right} =:leftVert xrightVert _{infty}$
for any $left(x_{1},cdots,x_{n}right)in R^{n}$. Now
do we have $lim_{prightarrow0}leftVert left(x_{1},cdots,x_{n}right)rightVert _{p}=leftVert left(x_{1},cdots,x_{n}right)rightVert _{0}:=mbox{cardinality}left{ x_{i}:x_{i}neq0right} $?