Escape velocity questions

This topic really screwed with my head in school and my teacher could never answer my questions so I thought I would ask here. I will break it down with mathematics and then logic.

Let’s assume we only have 1 body which is earth and nothing else exist.

We can work out the escape work done using this formula where $m = 1$:

Then transfer work done into velocity:

And you get the answer 11182.36 meters per second, which is correct for earth escape velocity. So the mathematics check out. The reason I’m including this is because I’m going to use the same variables(mainly ∞).

The definition for gravity is that it’s always attractive and is infinite. So the change in velocity(Because we’re escaping it is presumed it’s moving away from the mass) is always decreasing, or $dfrac{dy}{dx}$ is $downarrow$ and we can assume when $R = infty$, $dfrac{dy}{dx}$ is $0$. So going back to the equations above to reach infinite distance your going to need infinite time(that’s just the nature of how infinite works), but if we have infinite time and it’s multiplied by this negative $dfrac{dy}{dx}$ surely it’s impossible to escape a gravity field. Because no matter how small, any number times $infty$ is $infty$ and will result in an ∞ work done.

Hopefully you can understand that, and there is 2 separate mathematics ways of looking at it and both give different answers.

That was the mathematics part and that brings me onto the logic part. If gravity is infinite how can you ever escape?

I really hope someone can answer this because I literally can’t sleep because I think about this, maybe my understanding of $infty$ is just wrong. Also don’t try and use a calculator, it will get to a point where R is too large and the calculator doesn’t have enough bits so just rounds to $0$.