Sequence of functions converging in $L_(loc)^1(RR)$

[Camillo]

Build a sequence of functions $in L^1(RR)nn L^(oo)(RR)$ converging in $ L_(loc)^1(RR)$ respectively to the following functions:
a) $1$
b) $ x$
c) $|x|^(-1/2) $

[elgiovo]

Correct me if I go wrong: for the first two cases above a suitable sequence is

$u_k(x)=u(x)chi_([-k,k])$, $k in NN$, where $u(x)=1$ or $u(x)=x$.

These functions $in ccL^1(RR) capccL^oo(RR)$ and, with $k to oo$, they converge to $u(x)$ in $ccL_("loc")^1(RR)$.
In the third case, we'd better avoid the origin, so I believe a suitable sequence could be

$u_k(x)=u(x)(chi_([-k,-1/k])+chi_([1/k,k]))$, $k in N$, where $u(x)=|x|^(-1/2)$.

These functions are in $ccL_1 cap ccL^oo$ because $u(x)=|x|^(-1/2)$ belongs to $ccL_("loc")^1(ccOmega)$ $forall ccOmega ne RR$, and with $k to oo$ they converge to $u(x)$ in $ccL_("loc")^1(RR"/"{0})$.