- #1
kmm
- 188
- 15
In solving physics problems, I have often done some type of simplifying where I eliminated an x in the numerator and denominator, or eliminated some other terms. For example, maybe I have something like ## \frac {x} {x^2 + x} ## and I simplify this to ## \frac {1} {x+1} ##. Or I have something like ## \frac {x-1} {x^2-2x+1} \to \frac {x-1} {(x-1)^2} \to \frac {1} {x-1} ##. Typically, this is something I've taken for granted and done without much thought. But it dawned on me that in the first example, the original function is undefined at x=0, and the simplified version is one, at x=0 but has asymptotic behavior around x=-1. Also to get to this simplified version, by factoring out an x in the numerator and denominator, I had to assume ## \frac {x} {x} =1 ##. But ## \frac {x} {x} ## is undefined at x=0, so how can I simply assume it is 1? For the second example, ## \frac {x-1} {x^2-2x+1} ## is undefined at x=1, but ## \frac {1} {x-1} ## goes to infinity at x=1. And to go from ## \frac {x-1} {(x-1)^2} \to \frac {1} {x-1} ##, I again had to assume that ## \frac {x-1} {x-1} = 1 ##, but this is undefined at x=1. I'm not sure how I am supposed to be thinking about all of this. Is there some sort of redefining of process of these functions for these situations? I'm aware of L'Hostpitals Rule for indeterminate forms, but it has never occurred to me that I should use it in these situations where I'm simplifying a basic function.