Line that approaches zero but doesn't touch it?

Question

I'm sure this concept has been discussed before, but I couldn't find any information about it. Due to the laws of uncountable sets, no matter how small a number is (assuming it's a decimal), you can always make it smaller. You can always add another digit. This means there are an uncountably infinite set of numbers between 1 and 0, and could reliably count smaller and smaller numbers forever. In this fashion, could a totally straight line's y-value approach zero forever, but never touch it? Meaning, even if it continued for infinity, its y-values never reach 0 or become negative? What would the slope-intercept form of this line look like?

Answer 1

The word you are looking for is asymptote.

A straight line cannot ever be an asymptote.

The easiest example of an asymptote is y = 1/x as x goes to infinity.

Another more complicated example is (x+5)/(1+e^-x), which is asymptotic to y=x+5.