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(abbreviated) We show that there is a limit relation between the black holemass and the width at the half maximum of H$\beta$ for active galactic nuclei(AGNs) with super-Eddington accretion rates. When a black hole has asuper-Eddington accretion rate, the empirical relation of reverberation mappinghas two possible ways. First, it reduces to a relation between the black holemass and the size of the broad line region due to the photon trapping effectsinside the accretion disk. For the Kaspi et al.'s empirical reverberationrelation, we get the limit relation as $M_{\rm BH}=(2.9 - 12.6)\times10^6M_{\odot} (\upsilon_{\rm FWHM}/10^3{\rm km s^{-1}})^{6.67}$, called as theEddington limit. Second, the Eddington limit luminosity will be relaxed if thetrapped photons can escape from the magnetized super-Eddington accretion diskvia the photon bubble instability, and the size of the broad line region willbe enlarged according to the empirical reverberation relation, leading to arelatively narrow width of H$\beta$. We call this the Begelman limit.  Super-Eddington accretions in a sample composed of 164 AGNs have beensearched by this limit relation. We find there are a handful of objects locatebetween the Eddington and Begelman limit lines, they may be candidates ofsuper-Eddington accretors in a hybrid structure of photon trapping and photonbubble instability. The maximum width of H$\beta$ is in the reange of $(3.0 -3.8)\times 10^3$ km s$^{-1}$ for the maximum mass black holes withsuper-Eddington accretion rates among AGNs. We suggest that this limit relationis more reliable and convenient to test whether a source is super-Eddington anduseful to probe the structure of the super-Eddington accretion process.Comment: 5 pages (emulateapj5.sty), 1 figure. Astronomical Journal, 125 (June  Issue 2003) in pres

A Limit Relation between Black Hole Mass and H Width: Testing Super-Eddington Accretion in Active Galactic Nuclei