How do Euler's function work at denominator tending to zero?

You eliminate the factor by multiplying fuction by factor in denominator and calculate the function value at that point.

When is a rectangular parallelopiped maximized in volume?

When it is made a cube of side having the largest side of parallelopiped.

If y is a function of x What is dx/dy calculated ?

Express x in terms of y and take the derivative with respect to y.