If I have this

$$\displaystyle \lim\_{x\to\infty}\frac{3x}{6x}.$$
it is equivalent to this:

$$ \lim\_{x\to\infty}\frac{3}{6} = \frac{1}{2} $$

But couldn't I write the first expression as
$$ \frac{\lim\_{x\to\infty}3x}{\lim\_{x\to\infty}6x} = \frac{3(\infty)}{6(\infty)}$$

From my understanding that just becomes $\frac{\infty}{\infty}$ meaning there is no limit.

I'm clearly missing something here so to why these two don't converge to the same answer.
Can anyone help?