I need a thermometer for a variety of different applications. As such, it has to be fairly precise (plus/minus 2 degrees) at different temperatures. Do I have to worry that, if I buy a thermometer which has a large range (0 to 300 C, for example), that it won't be sensitive to small changes in temperature near 40 C?
There are several type of thermometer. Since you didn't specify exactly what you will be using for the thermometer It's hard to tell which type you are getting. However the two most likely kind you will use are bulb thermometer and bimetal thermometer.
The way bulb thermometer works is there is a liquid inside of a thin tube. Heat expands the liquid so the higher the heat the more it expands. The liquid is calibrated to expand to the markings on the thermometer.
However, the sensitivity of a thermometer does not depend on it's range but rather on several different factors:
A bimetal thermometer operates on the principle that different metals contract and expand to different degrees when exposed to temperature changes. Two strips made of different metallic materials are fused together as a spiral or rod. The strip will wind, unwind or bend as the temperature changes because one of the two materials in the strip will contract or expand to a greater degree than the other will. The free end of the strip is attached to a pointer that will indicate the temperature.
Once again because the dial moves base on the expansion (this time metal) which will stay constant, the range of the thermometer will not be affected.
Please note that over time these metal might degrade and lower the sensitivity of the thermometer but that is an issue of the longevity of the equipment.
Also note that is it harder to read the smaller changes by the human eyes if the range is larger since is scale is smaller but that has nothing to do with it's sensitivity.
If by chance you are refering to the uncertainty of a thermometer or how "incorrect" a thermometer can be, as Rumtscho mentioned, the uncertainty is usually given in the specs of the thermometer and you will need to do some research to find the one that will fit your needs. Just for example Tel-Tru boast an uncertainty of just +/- 1/10th degree Fahrenheit for it's products. You can usually find this sort of information on the product's website.
You don't have to worry about the range being too wide.
What you need is a thermometer which is sensitive enough to show a change even for a small temperature change. Sensitivity is a measure of its reaction to even a very small temperature change.
But I think there is another factor which goes along with sensitivity - the responsiveness. Even a very sensitive thermometer might take some time to show the temperature change due to its response being too slow.