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
The limit of an expression like 5x+3-x^2 as x approaches a value, like 2, is just the value the expression goes toward as x goes toward 2.
If you plug in 1.9, then 1.99, then 1.999, etc., you will see the values that the expression takes get closer and closer to 9. It just so happens that you can get 9 by simply plugging in x=2 to the expression as well. This shortcut works for “nice” expressions, which is why you will often see the limit disappear as something is plugged in for a value.
If you take the limit as h goes to 0 of 3x+h, you can just plug in h=0 in order to compute the limit as 3x because 3x+h is one of those “nice” expressions. In reality computing the limit is done by seeing what 3x+h moves toward as you plug in values that get closer and closer to 0 for h.
There is a more technical definition for calculating the limit, which is not usually taught in calculus the first time around, which defines exactly what people mean by “gets closer.”
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