There are several limits to the low frequency behaviour of spectrum analysers. Some are easy to intuit, some less so. Newer digital instruments are slightly different to the older analogue instruments in what they display.
One common limit is the high pass nature of the input DC blocking filter, which is generally no more than a series capacitor. That's quite simple. The filter might be -3dB at 9kHz, or it might be -1dB or even -0.1dB. It's fairly easy to see the level, and sort it out with calibration.
For the other limits, you need a block diagram for a typical spectrum analyser. This shows the simplified front end of a typical superheterodyne spectrum analyser.
This one goes up to 1GHz, and you can see from the diagram down to nominally DC.
The bandpass filter at 2GHz provides the selectivity. This is the filter that is switched to the various bandwidths that the analyser will provide, the 10MHz, 1MHz, 100kHz, 10kHz settings. For the larger bandwidths, it's a real switchable LC bandpass filter. For the smaller bandwidths, it's a system that uses further mix-down to lower IFs. In the modern digital analysers, the last IF is left fairly high frequency and wideband, and is digitised with FFT used for the final filters.
For a 0.5GHz input, the VTLO is set to 2.5GHz, producing a 2GHz IF.
For a 1GHz input, it's set to 3GHz, producing the same 2GHz IF.
For a nominally DC input, it's set to 2GHz.
On old analogue instruments, the sweep controlling the LO frequency could be set to sweep 'below' DC, for an input frequency below 0Hz. What you'd see on the display is the negative frequency for the first part of the sweep, a 'DC pip' at DC, and then the normal positive frequency range above that. Digital synthesiser based spectrum analysers tend to limit the display so you don't see either.
The input mixer, while it provides isolation between all its ports in theory, in practice 'leaks' some of the LO signal out to the 2GHz IF. This is the origin of the 'DC pip' mentioned above, aka 'LO leak'. The leakage can be quite severe. A not uncommon specification for RF spectrum analysers was for -40dBm signal at the input mixer, which tended to mean -50dBm to the IF. With a typical LO drive of +13dBm to +17dBm, you can see that even 60dB LO to IF isolation would still mean the DC pip was more than the expected signal. This sort of isolation needs active balance in the first mixer, and is still a struggle to achieve stably.
The LO leak causes two problems
1) Phase noise on the LO signal lifts the noise floor for input frequencies close to it, it reduces the dynamic range from the bottom. That's why 9kHz is an often seen lower limit. Just low enough to sneak into the audio range, high enough to get away from the worst of the LO phase noise.
2) The LO leak gets into the 2GHz IF, as a signal of similar size to the wanted signal. Generally the first element in the 2GHz IF is an LC filter, wide enough for the widest expected IF, perhaps 40MHz width. This filters out the LO leak for input frequencies above 20MHz. The distortion of the analyser is now controlled by the single tuned signal getting into the IF's amplifiers. This is generally the distortion that's specified for the instrument.
At input frequencies less than 20MHz, the LO leak now appears full strength at the IF amplifiers, reducing the signal handling of the amplifiers and increasing their distortion. This reduces the dynamic range from above. As mentioned above, the 2GHz IF is actually a system, using progressively lower frequency IFs and smaller bandwidth filters. There will therefore be a progressive degradation in dynamic range as the smaller bandwidth means that filters can no longer protect the detector from seeing the LO leak signal. Finally at DC, the LO gets through to the detectors, and the spectrum analyser can no longer distinguish the input signal.