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The question is: if lim(f(x)g(x)) doesn't exist, and lim(f(x)) = 1, and lim(g(x)) doesn't exist

then are we allowed to say that lim(f(x)g(x)) = lim(f(x)) * lim (g(x)) = 1 * lim(g(x)) = doesn't exist

or isn't that allowed in general

Thanks alot