# Amount of Error for a derivative

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?

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.