This statement is False.
To understand why, let’s break it down:
Given that
lim x→0 f(x) exists, it must approach some finite value L as x approaches 0. Similarly, if lim x→0 g(x) exists, it approaches some finite value M. However, just knowing these limits doesn’t give us enough information to conclude that the limit of the product f(x)g(x) as x approaches 0 is equal to 0.
In fact, if both L and M are non-zero, then, according to the limit product rule, the limit of the product is given by:
lim x→0 (f(x)g(x)) = lim x→0 f(x) * lim x→0 g(x) = L * M.
This product will only equal 0 if at least one of the limits (L or M) is 0, not automatically because both limits exist. For example, if L = 2 and M = 3 (both non-zero), then lim x→0 f(x)g(x) equals 6, not 0.
In conclusion, the statement is false because the product of the limits can yield a non-zero result if both individual limits are non-zero.