Measure an engine valve lift in a fixture
"Non-contact linear proximity sensors" are a special breed.
Digital is easy, Linear is not.

These include; laser proximity sensors, inductive , capacitive.
Linearity and resolution and range are critical  limitations. Inductive are most common while laser are most accurate. Most proximity sensors are "digital" output for some threshold, so ignore those. 

LVDS are linear by using a differential coil around a moving magnetic material to create a differential voltage,but smooth shafts cannot be detected and need a change in magnetic coupling by a small change in diameter equal to desired resolution.
 This may not be practical.

Cap Sensor from 6:10mm range. (min:max), 35Hz max
This does not have enough bandwidth for 3000 RPM = 50 Hz



e.g. http://www.ia.omron.com/products/family/1457/application.html



Optical reflector is non-linear but can be made linear for a small range.
After detailed review of specs. I would then choose the Laser reflectometer for displacement with &lt;0.01mm resolution. 

This is where it becomes a shopping question. Good luck.,