Can we create a 20 Hz EM wave? The electronic spectrum we study on our textbooks starts from radio waves of an order of kilohertz.
But is there a lower limit for EM radiation? Cant we create EM wave of any frequency from minus infinity to infinity? 
And if so why cant we transfer a 20 Hz - 20 Khz sound wave directly as an EM wave. Why should we use carrier waves on radio broadcasting
 A: 
But is there a lower limit for EM radiation? Cant we create EM wave of
  any frequency from minus infinity to infinity?

Infinite frequency is a bit high. Minus infinity does not make much physical sense. But, still, the range of possible wavelengths could be pretty wide: from subatomic to intergalactic.  

And if so why cant we transfer a 20 Hz - 20 Khz sound wave directly as
  an EM wave?

For efficient transmission, the size of antenna has to be comparable with the wavelength. For example, the length of a half wave dipole, tuned to  $20$kHz, would be $7500$ meters, which is impractical.

Why should we use carrier waves on radio broadcasting?

If we did not use carrier waves, all radio signals would be broadcast over the same band of audio frequencies and it would be impossible to select one particular station.
A: V.F. mentions the relationship between antenna size and frequency, but I'd like to make that more obvious.
This is an antenna that signals at 76 Hz. It is used to send messages to submerged submarines. The antenna is the part in the lower left, do you see what appears to be power lines running through the forest? Well that's one end of it, the other end is 14 miles away.
This is why we try to use high frequencies when we can. Low frequencies have their uses, but it doesn't make for a very practical cell phone.

A: Carrier waves are necessary to prevent interference. How would you distinguish a 1kHz sine signal from one source vs. another?
A: I don’t know why it is not accentuated in education, but the macroscopic source for electromagnetic radiation could be a thermic source or a modulated source.
In the case of a thermic source, subatomic particles - mostly the electrons - get randomly excited and emit photons of different frequencies and with a nearly constant amount over time.
In the case of a modulated source the electrons get excited periodically (a radio generator pushes and pulls them forward and backwards an antenna rod). Due to collisions of these electrons inside the rods metall they also emit photons of different frequencies. But the intensity of the radiation follows the macroscopic excitation of these electrons from the AC generator.
To generate any frequency is possible. The problem is the efficiency of the induced radiation. In the mentioned in the comment Wikipedia article it is stated, that “the antennas was about 1 megawatt, and they radiated about 2 watts of ELF radiation”.
One generate EM radiation even by waving a body. The body contains electrons and they get accelerated during the waving. Some amount of the kinetic energy gets realized by emission of photons from these accelerated electrons. BTW, thermodynamics is founded on the empirical knowledge, that any energy conversion - for example electric current into mechanical rotation - is accompanied by thermic loses.
