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Determine if the statement is true or false. If false, provide a counterexample.

(a) Given any real value k, if both lim f(x) and lim g(x) exist, then lim [f(x)+g(x)] must also exist.

User Leopinzon
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Final answer:

The statement is true, as the limit of a sum of two functions equals the sum of their individual limits, assuming both individual limits exist.

Step-by-step explanation:

The statement is true. When considering the limits of functions, if both lim f(x) and lim g(x) exist and are finite, then the limit of their sum lim [f(x) + g(x)] must also exist. This assertion can be justified through the properties of limits, specifically the limit laws. The limit of a sum equals the sum of the limits, provided that the limits of the summands exist. Mathematically, if lim x→c f(x) = L and lim x→c g(x) = M, then lim x→c [f(x) + g(x)] = L + M. Therefore, no counterexample is needed as the initial statement is true.

User Aaron He
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