Something is only infinite if the answer approaches infinity?

I was watching a maths video and the person claimed 1/0 was infinity, but I heard something is only infinite if it approaches infinity, despite the Google dictionary defining infinite as 'limitless or endless in space, extent, or size; impossible to measure or calculate'. No amount of 0s will ever reach 1 (zero is nothing!) So surely 1/0 is undefined, and does anyone know if it does need to approach infinity to classed as infinite?

You can't say anything *is* infinite because infinity is not a number. The best you can do is use the phrase 'tends towards' when talking about limits and so we get that $\dfrac{1}{x} \rightarrow \infty$ as $x \rightarrow 0$

Yes, so you can you say 1/0 tends to infinity, because it never approaches infinity?

This statement doesn't make sense.

For one, you used 1/0 which is an undefined quantity. It doesn't tend to anything.

Secondly, since you want to talk about 1/x instead as $x \rightarrow 0$, the phrases "tends to" and "approaches" are interchangeable. So your statement is the same as saying "1/x tends to infinity because it never tends to infinity" and clearly you can see how this doesn't make sense.

...

You can't say anything *is* infinite because infinity is not a number. The best you can do is use the phrase 'tends towards' when talking about limits and so we get that $\dfrac{1}{x} \rightarrow \infty$ as $x \rightarrow 0$

$1/x\rightarrow -\infty$ as $x\rightarrow 0^-$.