Radiance at a nearly perpendicular angle is almost infinite? As far as I understand, radiance is a measure of the "intensity" of light in a direction defined as a differential cone . It's measure is Flux per solid angle per projected area.  $L = \frac{d\Phi}{d\omega dA^\perp} = \frac{d\Phi}{d\omega dAcos(\theta)} $. Using this definition, it seems like as the $\theta$ angle increases, radiance increases, thus light is somehow more "intense" in a direction nearly perpendicular to the normal at point p. This contradicts my intuition.
Where am I interpreting this wrong?
It's only more intense if $d\phi$ remains constant. However, for light source with constant luminance the flux is a function of $cos(\theta)$, i.e. depends on the light's angle of incidence with the surface and cancels out the term.

Edit: I feel the need to clarify this a bit because you say that:


  thus light is somehow more "intense"


It's not that the "light is somehow more intense", but rather that the light has to be more intense to have same flux in grazing angles as when it's parallel to the surface. If you think of flux as particles hitting to this differential surface, the number of particles hitting the surface is smaller more perpendicular the light gets to the surface. So you have to throw more particles at the surface (i.e. increase the radiance) for the same flux until when the surface is completely perpendicular to the light, none of the particles hit the surface, thus the radiance goes to infinity.