Question

Determine whether the statement is true or lim x→7 f(x) = 0 and lim x→7 g(x) = 0, then lim x→7 f(x)g(x) does not exist.

a. True
b. False

Answer 1

The statement is false; the limit of the product of two functions, if each has a limit of zero as x approaches a specific value, is also zero, meaning that the limit does exist.

Step-by-step explanation:

The statement in the question suggests that if the limit of f(x) as x approaches 7 is 0, and the limit of g(x) as x approaches 7 is also 0, then the limit of the product f(x)g(x) as x approaches 7 does not exist. This statement is false. According to the properties of limits in calculus, if the limit of f(x) as x approaches a specific value is L and the limit of g(x) as x approaches the same value is M, then the limit of the product f(x)g(x) as x approaches that value is L times M. In this case, since both limits are 0, the limit of the product is 0 x 0, which is 0. Therefore, the limit does exist, and it is equal to 0.