To evaluate the statement, we need to consider the properties of limits in calculus. The question implies examining the behavior of two functions, f(x) and g(x), as x approaches 0.
Let’s denote:
- L1 = lim (x -> 0) f(x)
- L2 = lim (x -> 0) g(x)
The claim suggests that if both L1 and L2 exist, then:
- lim (x -> 0) (f(x) * g(x)) = 0
While it’s correct that if both limits exist, we can often find the limit of the product as:
lim (x -> 0) (f(x) * g(x)) = L1 * L2. In this case, the result will be 0 only if either L1 = 0 or L2 = 0. Otherwise, the product will converge to the multiplication of the two non-zero limits.
For instance:
- If f(x) = x and g(x) = 1, then:
- lim (x -> 0) f(x) = 0
- lim (x -> 0) g(x) = 1
- lim (x -> 0) (f(x) * g(x)) = 0 * 1 = 0
- If f(x) = 1 and g(x) = 1, then:
- lim (x -> 0) f(x) = 1
- lim (x -> 0) g(x) = 1
- lim (x -> 0) (f(x) * g(x)) = 1 * 1 = 1
In conclusion, the statement is False. The limit of the product of two functions does not automatically equal zero when the limits of the individual functions approach zero. Instead, it is necessary to analyze the values of the individual limits. Therefore, we can assert that:
False: lim (x -> 0) (f(x) * g(x)) does not equal 0 unless one of the limits is indeed 0.