Let  $f:(1,\infty )\to (0,\infty )$  be a continuous decreasing function with $\underset{x\to \infty }{\lim }\frac{f\left(4x\right)}{f\left(8x\right)}=1$ .  Then, $\underset{x\to \infty }{\lim }\frac{f\left(6x\right)}{f\left(8x\right)}$   is equal to _________

$f\left(x\right)$  is continuous decreasing positive function on  $(1,\infty )$ 
 $$\begin{gathered} \therefore 0<f\left(8x\right)<f\left(6x\right)<f\left(4x\right) \\ \therefore 0<1<\frac{f\left(6x\right)}{f\left(8x\right)}<\frac{f\left(4x\right)}{f\left(8x\right)} \end{gathered}$$
$\therefore$  By sand witch theorem, $\underset{x\to \infty }{\lim }\frac{f\left(6x\right)}{f\left(8x\right)}=1$