Question

If both $\underset{x\to a}{lim}f\left(x\right)$ and $\underset{x\to a}{lim}g\left(x\right)$ and exist finitely and $\underset{x\to a}{lim}g\left(x\right)=0$, then
$\underset{x\to a}{lim}\frac{f\left(x\right)}{g\left(x\right)}=\frac{\underset{x\to a}{lim}f\left(x\right)}{\underset{x\to a}{lim}g\left(x\right)}$

• A. True
• B. False

Answer 1

The correct option is B

False

The given result is true for all the cases except when $\underset{x\to a}{lim}g\left(x\right)$ is zero.
So the result
$\underset{x\to a}{lim}\frac{f\left(x\right)}{g\left(x\right)}=\frac{\underset{x\to a}{lim}f\left(x\right)}{\underset{x\to a}{lim}g\left(x\right)}$
is applicable only when $\underset{x\to a}{lim}f\left(x\right)$ and $\underset{x\to a}{lim}g\left(x\right)$ exist finitely, and $\underset{x\to a}{lim}g\left(x\right)\ne 0$.

Thus the given statement is wrong.