If limx→a f(x)g(x) exists, then

If $lim_{x \to a} (\frac{f (x)}{g (x)})$ exists, then

A
both $lim_{x \to g} f (x) and lim_{x \to 0} g (x)$ must exist
B
$lim_{x \to a} f (x)$ need not exist but $lim_{x \to a} g (x)$ exists
C
neither $lim_{x \to a} f (x) nor lim_{x \to a} g (x)$ may exist
D
$lim_{x \to a} f (x)$ exists but $lim_{x \to a} g (x)$ need not exist

Solution:

If $f (x) = \sin (\frac{1}{x}) and g (x) = \frac{1}{x},$ then both $\lim_{x \to 0} f (x)$

and $lim_{x \to 0} g (x)$ do not exist but $lim_{x \to 0} \frac{f (x)}{g (x)} = 0$ exists.

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