Is it possible that limx→af(x) doesnt exist, limx→ag(x) exists e limx→a(f(x)⋅g(x)) exists? true or false?

1. Consider the function f(x) = \sin\left(\frac{1}{x-a}\right), which has no limit as x approaches a because it oscillates between -1 and 1. Therefore \lim_{x \to a} f(x) doesn't exist.
2. Let g(x) = 0 for all x. This means \lim_{x \to a} g(x) = 0, which is defined and exists.
3. The product f(x) \cdot g(x) = \sin\left(\frac{1}{x-a}\right) \cdot 0 = 0 for all x except at a.
4. Thus, \lim_{x \to a} (f(x) \cdot g(x)) = \lim_{x \to a} 0 = 0, which exists.
5. Therefore, it is true that \lim_{x \to a} f(x) doesn't exist, \lim_{x \to a} g(x) exists, and \lim_{x \to a} (f(x) \cdot g(x)) exists.

The answer is: True.