TECHNICAL ALERT ! the following will probably be gobbldygook to anyone without good (university level) knowledge of electronics.
"Resolution is 5 hz, not 5 khz. Oops......" - W0ttm
5Hz is not bad for a 80s/90s kit .. LC oscillator and a ton of IC's !? .. 0.9 to 31MHz with 5Hz resolution - frequency set on thumbwheels ... quite acceptable!
My HP-3325A synthesisers (I have 2) have been some of the most useful kit I own - covers 1 uHz up to 20 MHz with resolution from 1 uHz (highest) to 1mHz (.001 Hz) at 20MHZ .. They are just incredible machines - and great for simulating theremin oscillators when developing mixers / waveshapers etc.. You may find the service manual for these useful, as it goes into great depth about how things work: http://cp.literature.agilent.com/litweb/pdf/03325-90002.pdf
It may be that your LC oscillator is a VCO in a PLL system, with phase error CV controlling a varicap.
Sadly, these instruments are huge heavy bulky bits of kit.. Mostly I dont need better than 1Hz resolution, so a tiny PC driven board like this with 0 to 30Mhz coverage and worst case 0.3Hz resolution, is an ideal solution - two of these (at £5 each) would actually do most of what I use my big synthesisers for ..
Fred.