Is the function $f( x)=1/|x|^{1/2}$ Lipschitz continuous? Is the function $f( x)=1/|x|^{1/2}$ Lipschitz continuous near $0$? If yes, find a constant for some interval containing $0$
I  think the answer is yes since I can find $L=1$ that satisfies Lipschitz continuity criteria in a interval close to zero,am I right?

At zero the function $f(x)$ is not Lipschitz continuous any more, it is not even simply continuous. Close to zero - yes, but not in zero.
For every constant $C$ you can find $x_1$ and $x_2$ such that $|f(x_1)-f(x_2)| > C|x_1-x_2|$. For example if $C > 1$ you can take $x_1 = \frac{1}{C}$ and $x_2 = \frac{1}{2C}$.
