How to prove lim(f(x)+g(x))= a+b as x->x0 and lim f(x)=a, lim g(x)=b? Could someone help me?
proofs of various limit properties here ...
Pauls Online Notes : Calculus I - Proof of Various Limit Properties
Look at |f(x)+g(x)- (a+b)|= |(f(x)-a)+(g(x)-b)| and use the fact that . In order to make , we have to make sure that |f(x)-a| and |g(x)-b| add up to something less than . One way to do that is to make sure that |f(x)-a| and |g(x)-b| each are less than . Since , there exist some such that if [tex]|x- x_0|< \delta_1[\math], . Since , there exist some such that if , .
To be sure that both of those happen, choose to be less than or equal to the smaller of and . That way, if , both and are true.