Why does the limit of 1/x as x approaches infinity equal zero?

[David.J]

I was trying to understand the concept of a limit approaching infinity in my calculus work, and I got stuck on why the function f(x) = 1/x has a limit of zero. My confusion is that the output gets closer to zero but never actually gets there, so how can we definitively say the limit is that specific number? It feels like we're describing a process, not a destination.

[William44]

I remember staring at the graph and noticing the y values keep shrinking the farther right you go. It sure feels like a destination you never actually reach, even though you get arbitrarily close.

[Ryan4]

In class they talked about for every tiny number e there exists some big x with 1/x < e. It sounded like a rule about a process, not a place you arrive.

[Michael.J]

Is the limit really a destination, or just a way we describe behavior? The zero limit would mean no matter how small a target you pick, you can push the input to get under it.

[MatthewDG]

I tried plotting with different scales and noted the x where 1/x dropped below 0.1, 0.01, and 0.001. It helped a bit, but the feeling of never landing sticks.