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Let f: D -> R and g: D->R where D is an open interval that contains the point c and let f and g be functions that are defined on D except possibly at point c.

Suppose that limit_(x->c) f(x) exists and that limit_(x->c) g(x) does not exist.

Define the function h(x) = f(x) + g(x).

Prove that limit_(x->c) h(x) does not exist.

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Solution Summary

This posting explores the concept of arbitrary epsilon.

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Suppose, the limit of h(x) as x approaches c does exist and equals L. This means that for every epsilon>0 we can find delta>0 such that for all x satisfying |x-c|<delta we have:
|h(x)-L|<epsilon
Also, suppose that the limit of f(x) as x approaches c, is M.
Then, ...

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