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Exercises 45-51. True or false? Justify your answers. 45. If lim _{x → c}[f(x)+g(x)] exists but lim _{x → c} f(x) does not exist, then lim _{x → c} g(x)

Answer :

False. The statement is false because the existence of the limit of the sum of two functions does not provide any information about the individual functions' limits.

If the limit of the sum of two functions, f(x) + g(x), exists at a given point c, but the limit of f(x) does not exist at that point, we cannot determine whether the limit of g(x) exists or not. The existence of the limit of the sum does not provide any information about the individual functions involved.

To understand why this statement is false, consider the following scenario: let f(x) be a function that oscillates indefinitely as x approaches c, such as f(x) = sin(1/x), and let g(x) be a constant function.

In this case, the limit of f(x) as x approaches c does not exist because the function keeps oscillating. However, the limit of the sum, f(x) + g(x), might still exist depending on the constant value of g(x).

For example, if g(x) is a constant value of 0, then the limit of f(x) + g(x) will be 0, even though the limit of f(x) does not exist. On the other hand, if g(x) is a non-zero constant, the limit of f(x) + g(x) will not exist since f(x) oscillates.

Therefore, the existence or non-existence of the limit of one function does not determine the existence or non-existence of the limit of another function when they are summed together.

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