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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ELI5: imagine you have a cup of hot choclate and your mom says “when you finish your hot chocolate you have to go to bed”. Now, you are smart and you don’t want to go to bed, so everytime you take a sip, you make sure that you leave some hot chocolate in the cup. You can drag this out forever and never finish the hot chocolate, so the amount left is always more than nothing. Now imagine that mom gets annoyed and starts looking how much is left in the cup with a ruler. The longer you do it, the closer to 0 the lines of the ruler will become that stick out of the chocolate that’s left. Eventually she could draw new lines of the ruler, ever closer to 0 and be sure that at some point the amount of hot chocolate is less than that. In fact, she could pick or add any line to the ruler, anywhere, and be sure that after you play this game for some time this line on the ruler will forever stick out of the hot chocolate left. So in short: The hot chocolate will never be empty, because you always leave some in the cup. But you can pick any level, however close to empty, and you know that after some sip the hot chocolate will always be less than that. So the limit of the hot chocolate is empty but it will still never be empty.
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