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
While it is not the correct definition of a limit, in most everyday cases of using limits to deal with functions, you’re basically asking “What is the right value of this function to fill in at this point so that it becomes continuous (i.e. its graph becomes connected)? Or is there even one?”.
Suppose you’re computing something like lim_(x→1) x+1. When you just plug in x=1 to get 2, the often unstated logic behind this is roughly as follows: x+1 is already a continuous function, and it already has a value at x=1, which is 2. Hence, the right fill-in to make it continuous is just the original value, 2.
Suppose you’re computing something like lim_(x→1) (x²-1)/(x-1). Now you originally have no value at x=1, because that gives you 0/0. But you notice everywhere other than x=1, (x²-1)/(x-1) and x+1 are in fact exactly the same. So they should certainly approach the same value at x=1. And then the substitution of x=1 into x+1 is justified with the same logic as above: x+1 is already a continuous function, so the correct “fill-in” it approaches as x approaches 1 is just the original value at x=1.
(Again: if you start getting technical, the above explanation is not completely correct. It for instance assumes that your function is nice and continuous near the point you’re taking a limit at, and it applies somewhat circular logic in explaining limits through continuity while continuity is formally defined through limits. But it should at least somewhat demystify the precise logic behind these kind of computational manipulations that are typically done with limits.)
Latest Answers