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If lim x→5 f(x) = 1 and lim x→5 g(x) = 0, then lim x→5 f(x) g(x) does not exist.

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Final answer:

The product of the limits of two functions is equal to the product of their individual limits, provided the individual limits exist. Hence, lim x→5 f(x)g(x) does exist, and it is equal to 0.

Step-by-step explanation:

The question provided involves understanding the concept of limits in calculus, a fundamental concept used to analyze how functions behave as inputs approach a certain value. In this case, we are given that lim x→5 f(x) = 1 and lim x→5 g(x) = 0. From these limits, we can infer the behavior of the product of these functions as x approaches 5.

According to the properties of limits, if the limit of f(x) as x approaches a number is a finite value, and the limit of g(x) as x approaches that same number is 0, then the limit of the product of these functions will be the product of their limits. Hence, lim x→5 f(x)g(x) = lim x→5 f(x) × lim x→5 g(x) = 1 × 0 = 0. Therefore, the initial claim that "lim x→5 f(x)g(x) does not exist" is not correct; the limit does exist, and it is zero.

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