TITLE: Calculus Paradox. I mean, what's wrong with what I think?
QUESTION [7 upvotes]: Is not calculus based on the paradox that the closest point to a point A is a distinct point B which is the point A itself?
For example, if we consider the limit,
$$ \lim_{x\to2} \frac{x^2-4}{x-2} $$ 
It's evaluated by first cancelling out the $(x-2)$ common factor and then we substitute the value $2$ in the function of $x$. It's like, at first, we're considering that $x$ is nearly equal to $2$ but not $2$, but then we substitute the value of $2$. So, what's happening here? Similarly, in derivatives we're considering the tangent to a curve which intersects the curve at one distinct point. That's how we get the exact slope. But, at times, we're considering a point $A$ which is close to point $B$ (and point $B$ and point $A$ are different). But, we know that the concept of derivatives is legit by experimental evidence. So, how does it all actually work?

REPLY [3 votes]: Until the early 1800s, you would have been essentially correct in your understanding. Calculus was considered an intuitive technique rather than a rigorous mathematical theory. Cauchy changed that by establishing a rigorous theory for it.