suppose that f : (a,b)\{c} ----> real numbers is a function such that lim (x--->c+) {f(x)} and lim (x--->c-) {f(x)} both exist and are equal to a common value l. how can we prove that lim (x--->c) {f(x)} exists and that it equals l?
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If and then one of these is true, . The first is involved with the limit from the left the other the limit from the right.
Let be given. By assumption, we can find a such that for or , we have that . Set . Then for , it follows that .
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