How to find the points that the function is not analytic?

[gohkgohkgohk]

For example,

f(x)=(x+2)/((x-1)*(x-2))

We know that f(x) is not analytic for x=1 and x=2 (from our eye) ,
but do there exist a general method to find out these points?

Thank you

 

[HallsofIvy]

For rational functions, of course, the function is not analytic if and only if the denominator is 0 (where the function is not defined) but there is no general way of determining where a function is not analytic.