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I am reviewing for a test in my Calculus III class and I ran into a problem. I need to prove that $\lim_{(x,y)\to(a,b)} x+y=a+b$ using the formal definition of a limit.

The definition given is:
Let $f$ be defined on the interior of a circle centered at the point $(a, b)$, except possibly at $(a, b)$ itself. We say that $lim_{ (x,y)→(a,b)}f (x, y) = L$ if for every $ε > 0$ there exists a $δ > 0$ such that $| f (x, y) − L| < ε$ whenever $0 < \sqrt{(x − a)^2 + (y − b)^2} < δ$.

Can someone please help me? Thank you.

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up vote 1 down vote accepted

Assume that $ε > 0$ is given. Note that $0 < \sqrt{(x − a)^2 + (y − b)^2} < δ$ implies $|x-a| < \delta$ and $|y-b|<\delta$. So if we take $\delta=\frac{\epsilon}{2}$, then $|(x+y) - (a+b)| \leq | x-a| + |y-b| < 2\delta=\epsilon$.

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