Suppose that f and g are real valued functions defined on some interval (a,b) containing the point . Suppose also that and . Prove that if for all (for some ). Then . In this case is it always true that A<B.
For the first part my proof so far consists of a contradiction. I have assumed A>B and then proceded to write out the definitions of limit tending to with and so on. However I can't seem to find a contradiction and was wondering if someone could give me the first few lines I need.
I also have no idea for the second part. Thanks!
There might be an even simpler example, but since the limit has to be taken at an interior point of an interval I could only think of doing something like f(x)=0, g(x)=|x| if x is nonzero and g(0)=1. These are both defined on (-1,1) and g>f there. Now take limits as you approach 0.