Question
If $$\mathop {\lim }\limits_{x \to a} \left[ {\frac{{f\left( x \right)}}{{g\left( x \right)}}} \right]$$ exist, then which one of the following correct ?
A.
Both $$\mathop {\lim }\limits_{x \to a} f\left( x \right)$$ and $$\mathop {\lim }\limits_{x \to a} g\left( x \right)$$ must exist
B.
$$\mathop {\lim }\limits_{x \to a} f\left( x \right)$$ need not exist but $$\mathop {\lim }\limits_{x \to a} g\left( x \right)$$ must exist
C.
Both $$\mathop {\lim }\limits_{x \to a} f\left( x \right)$$ and $$\mathop {\lim }\limits_{x \to a} g\left( x \right)$$ need not exist
D.
none of these
Answer :
none of these
Solution :
$$\eqalign{
& f\left( x \right) = x,\,\,g\left( x \right) = \frac{1}{x} \cr
& \mathop {\lim }\limits_{x \to 0} f\left( x \right) = 0,\,\,\mathop {\lim }\limits_{x \to 0} g\left( x \right) = {\text{does not exist}} \cr
& {\text{But }}\mathop {\lim }\limits_{x \to 0} \left[ {\frac{{f\left( x \right)}}{{g\left( x \right)}}} \right] = \mathop {\lim }\limits_{x \to 0} \left[ {{x^2}} \right] = 0 \cr} $$
Hence, none of these is only true option.