Question

If $\underset{x\to a}{lim}\left\{\frac{f\left(x\right)}{g\left(x\right)}\right\}$ exists, then

• A. both $\underset{x\to a}{lim}f\left(x\right)$ and $\underset{x\to a}{lim}g\left(x\right)$ exist
• B. $\underset{x\to a}{lim}f\left(x\right)$ need not exist but $\underset{x\to a}{lim}f\left(g\right)$ exist
• C. neither $\underset{x\to a}{lim}f\left(x\right)$ nor $\underset{x\to a}{lim}g\left(x\right)$ may exist
• D. $\underset{x\to a}{lim}f\left(x\right)$ exists but $\underset{x\to a}{lim}g\left(x\right)$ need not exist

Answer 1

The correct option is A both $\underset{x\to a}{lim}f\left(x\right)$ and $\underset{x\to a}{lim}g\left(x\right)$ exist

If $\underset{x\to a}{lim}\left\{\frac{f\left(x\right)}{g\left(x\right)}\right\}=\frac{\left\{\underset{x\to a}{lim}f\left(x\right)\right\}}{\left\{\underset{x\to a}{lim}g\left(x\right)\right\}}$

So, both numerator and denominator should exist so as to $\underset{x\to a}{lim}\left\{\frac{f\left(x\right)}{g\left(x\right)}\right\}$ exist.