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?

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- Nov 22nd 2009, 08:18 AManteroHow to prove? (limit)
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?

- Nov 22nd 2009, 08:49 AMskeeter
proofs of various limit properties here ...

Pauls Online Notes : Calculus I - Proof of Various Limit Properties - Nov 22nd 2009, 09:53 AMantero
Thank you. But I didn't understand how to find out the value of delta?

- Nov 22nd 2009, 10:12 AMHallsofIvy
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.