Question

If $\underset{x\to a}{lim}f\left(x\right)g\left(x\right)$ exists, then does it imply that $\underset{x\to a}{lim}f\left(x\right)$ and $\underset{x\to a}{lim}g\left(x\right)$ also exist? Yes =5 No=7

Answer 1

No. Let us choose $f\left(x\right)=x,g\left(x\right)=\frac{1}{x}$
$\underset{x\to 0}{lim}f\left(x\right)g\left(x\right)=\underset{x\to 0}{lim}1=1.$
$\underset{x\to 0}{lim}f\left(x\right)=\underset{x\to 0}{lim}x=0.$ Limit exists.But $\underset{x\to 0}{lim}g\left(x\right)=\underset{x\to 0}{lim}\frac{1}{x}.$ This limit does not exist
as $R.H.L.=\infty ,L.H.L.=-\infty .$