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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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.
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