A) Both \[\underset{x\to a}{\mathop{\lim }}\,f(x)\] and \[\underset{x\to a}{\mathop{\lim }}\,g(x)\] must exist
B) \[\underset{x\to a}{\mathop{\lim }}\,f(x)\] need not exist but \[\underset{x\to a}{\mathop{\lim }}\,g(x)\] must exist
C) Both \[\underset{x\to a}{\mathop{\lim }}\,f(x)\] and \[\underset{x\to a}{\mathop{\lim }}\,g(x)\] need not exist
D) None of these
Correct Answer: D
Solution :
[d] \[f(x)=x,g(x)=\frac{1}{x}\] \[\underset{x\to 0}{\mathop{\lim }}\,f(x)=0,\underset{x\to 0}{\mathop{\lim }}\,g(x)=\] does not exist But \[\underset{x\to 0}{\mathop{\lim }}\,\left[ \frac{f(x)}{g(x)} \right]=\underset{x\to 0}{\mathop{\lim }}\,\left[ {{x}^{2}} \right]=0\] Hence, none of these is only true option.
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