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Data from {\it Chandra} observations of thirty nearby galaxies were analyzedand 365 X-ray point sources were chosen whose spectra were not contaminated byexcessive diffuse emission and not affected by photon pile up. The spectra ofthese sources were fitted using two spectral models (an absorbed power-law anda disk blackbody) to ascertain the dependence of estimated parameters on thespectral model used. It was found that the cumulative luminosity functiondepends on the choice of the spectral model, especially for luminosities $>10^{40}$ ergs/s. In accordance with previous results, a large number ($\sim80$) of the sources have luminosities $> 10^{39}$ ergs/s (Ultra-Luminous X-raysources) with indistinguishable average spectral parameters (inner disktemperature $\sim 1$ keV and/or photon index $\Gamma \sim 2$) with those of thelower luminosities ones. After considering foreground stars and knownbackground AGN,we identify four sources whose minimum luminosity exceed$10^{40}$ ergs/s, and call them Extremely Luminous X-ray sources (ELX). Thespectra of these sources are in general better represented by the disk blackbody model than the power-law one. These ELX can be grouped into two distinctspectral classes. Two of them have an inner disk temperature of $< 0.5$ keV andhence are called ``supersoft'' ELX, while the other two have temperatures$\gtrsim 1.3$ keV and are called ``hard'' ELX. The estimated inner disktemperatures of the supersoft ELX are compatible with the hypothesis that theyharbor intermediate size black holes, which are accreting at $\sim 0.5$ timestheir Eddington Luminosity. The radiative mechanism for hard ELX, seems to beInverse Comptonization, which in contrast to standard black holes systems, isprobably saturated.

The dependence of the estimated luminosities of ULX on spectral models