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.
In: 8
So when you look at a limit, if you were to “read it aloud” you’d say, for example, “The limit as x approaches 0 of….”
The key being “approaches”. The limit equals a value because the limit is telling you the value that a function is approaching. The function is equal to that value – it’s approaching it.
This is a great question and one that prompted much thought during the 18th and 19th centuries. Essentially you want to know why should we believe in limits? It turns out that the real numbers are a special type of space called a complete metric space, which means that if there is a Cauchy sequence (something that gets arbitrarily closer and closer together), then the limit of the sequence exists as part of the space. When you talk about the limit of a sequence you’re not talking about any given point in the sequence – you’re talking about the point that you get arbitrarily close to. And since it’s a complete metric space, the limit also must be a real number.
In other words, the limit is not a process, it’s a target. A sequence that converges to the limit is not the limit itself.
Latest Answers