I only ever encountered the limit while learning derivation by first principles in calculus. I understood all the theory behind first principles, but we were never told what happens to the limit h -> 0. Our teacher just said that it goes away after we divide by h, and that’s all I got.
I understand that the limit h -> 0 represents the gap between x and (x + h) getting smaller and smaller. But how does this gap disappear at the end? From searching online I’ve learned that limits are not *equality*, h never *equals* zero, it just gets closer and closer to it. But then why does it equal zero at the end? How is h -> 0 no longer intrinsic to f'(x)? This might be a dumb question but it has stumped me for ages now.
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Yeah, limits tend to get hand waved a bit in elementary calculus, especially in high school. If you get further into math, you’ll come across a more rigorous definition, which I think helps clarify things.
Let’s consider the function [f(x) = sin(x) / x](https://www.wolframalpha.com/input/?i=plot+sin%28x%29%2Fx). Looking at the graph, it seems like f(0) = 1, but if you plug in 0 you get 0 / 0 which is undefined. And indeed, the function is defined for all x except 0, and skipping ahead a bit, lim[x->0] f(x) = 1. If you’ve learned L’Hôpital’s rule, a quick application of it confirms this.
But if we back up a bit, what’s a general way we could convince ourselves that this limit should be 1? The idea is to think about how close we can get to 1, and make a little game out of it. Let’s say I challenge you to find a value of x such that f(x) is within 0.01 of 1. Can you do it? It turns out, you can, x=0.000001 does it. What about getting f(x) within 0.0000001 of 1? Sure, just make x=0.000000000000000001. No matter how close I challenge you to get to 1, you can find an x that does it, and you have to get close to 0 to find it. This is the [epsilon-delta](https://en.wikipedia.org/wiki/Limit_of_a_function#Functions_of_a_single_variable) definition.
The idea works the same with lim[x->infinity] 1/x = 0. You can’t “plug in” infinity, but however close to 0 I challenge you to get, you can find an x that achieves it, and you have to get arbitrarily large to do so.
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