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Answer :
The statement is true, and the counterexample will show the converse is false.
In mathematics, when the limits of two functions, f(x) and g(x), exist as x approaches a certain value (let's say 'a'), it does not necessarily imply that the limit of their product, f(x) * g(x), also exists at 'a'.
However, if the limits of f(x) and g(x) both exist, then the limit of their product f(x) * g(x) does exist. This is due to the fact that the product of two continuous functions is also continuous.
To illustrate the first part, consider the following counterexample: let f(x) = 1/x and g(x) = x. As x approaches 0, the limits of both f(x) and g(x) exist: lim (1/x) = ∞ and lim (x) = 0.
However, the limit of their product lim (1/x * x) = lim (1) as x approaches 0, which is a constant, not infinity. Hence, the converse does not hold in this case.
On the other hand, when both f(x) and g(x) have existing limits, the limit of their product can be shown to exist. This can be demonstrated by employing the limit laws and continuity properties of functions.
As long as the limits of f(x) and g(x) exist, the limit of their product will also exist, and it will be equal to the product of their individual limits.
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The statement is true, and the counterexample will show the converse is false.
In mathematics, when the limits of two functions, f(x) and g(x), exist as x approaches a certain value (let's say 'a'), it does not necessarily imply that the limit of their product, f(x) * g(x), also exists at 'a'.
However, if the limits of f(x) and g(x) both exist, then the limit of their product f(x) * g(x) does exist. This is due to the fact that the product of two continuous functions is also continuous.
To illustrate the first part, consider the following counterexample: let f(x) = 1/x and g(x) = x. As x approaches 0, the limits of both f(x) and g(x) exist: lim (1/x) = ∞ and lim (x) = 0.
However, the limit of their product lim (1/x * x) = lim (1) as x approaches 0, which is a constant, not infinity. Hence, the converse does not hold in this case.
On the other hand, when both f(x) and g(x) have existing limits, the limit of their product can be shown to exist. This can be demonstrated by employing the limit laws and continuity properties of functions.
As long as the limits of f(x) and g(x) exist, the limit of their product will also exist, and it will be equal to the product of their individual limits.
Learn more about converse.
brainly.com/question/31918837
#SPJ11
