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Although $L^{1}\sb {loc}$ does not contain the Dirac distribution, it may be useful to distinguish $L^{1}\sb {loc}$-distributions from say distributions represented by Radon measures. Your question is interesting, because it is definitely important to understand examples of distributions. That said, perhaps the motivation for distributions is equally important. Distributions help us take weak derivatives. The definition of a derivative of a distribution is motivated by Integration by Parts.

As you may know, often many mathematicians are more interested in working with specific distributions such as those in Sobolev spaces such as $W^{k,p}(U)$ ($1\leq p\leq \infty$), $BV(U)$ (integrable functions whose first order (weak) derivatives are signed measures with finite variation), or even tempered distributions. Then there are distributions like $T(\phi):=\sum\sb {k=1}^{\infty}\int\sb {0}^{1}\frac{\phi(x)sin(k\pi x)}{x}dx$ for $\phi\in C\sb {c}^{\infty}((0,1))$. I guess the point is, be careful not to think that all distributions somehow behaving the same way.