Final Answer:
The product of the limits of two functions, x→a f(x) and x→a g(x), exists despite the individual limits of f(x) and g(x) not existing at x=a.
Explanation:
When evaluating the limit of the product of two functions, f(x) and g(x), at x=a, it is possible for the limit to exist even if the individual limits of f(x) and g(x) do not. This occurrence doesn't violate mathematical principles due to the interplay between the functions.
The product limit may stabilize or converge despite the individual functions exhibiting behavior that prevents their own limits from existing at the same point. Such a scenario typically arises when the oscillations or the manner in which the functions behave cancel each other out when combined. This cancellation allows for a defined limit of their product.
Understanding the interaction between the functions is crucial in grasping why the limit of their product can exist while the individual limits do not.
This phenomenon emphasizes the complexity of function behavior near a specific point and how different functions can interact to yield surprising results.