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I just learned what a Derivative is in my high school calculus class.

The formula, as I learned it, is:

$\lim\limits_{h \to 0} \dfrac{f(x+h)-f(x)}{h} $

However, the way I learned Limits was that they were approaching a number, and would never actually equal to the number the limit would resolve to. The way I comprehend this is that there is a small error (or number) between the limit of the function and the actual evaluation of the function.

Therefore, does that mean derivatives also have a slight error, and is not completely $100\%$ exact?

My teacher told me to google it, but I couldn't find anything that goes into detail about this.

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Something being a limit value says that we can make the error as small as you want. That's the language for any $\epsilon>0$ ($\epsilon$ is the tiny error in the output), we can find a $\delta >0$ such that as long as we are no more than $\delta$ away from $x$ on our input, our output will be within $\epsilon$ of our limit.

In other words, by zooming in to a small region on the x axis, the variation/error on the y axis gets as small as you want it. So while it may never actually equal that value, we can use the number effectively as an "instantaneous" value at that point.

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