Calculus Paradox. I mean, what's wrong with what I think? 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?
 A: 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.
A: What is happening is that the two functions
$$
x\mapsto \frac{x^2-4}{x-2} \text{ and } x\mapsto x+2
$$
both have the same values at all inputs $x$ except $x=2$, and the first function is undefined at $x=2$ and the second is not only defined but continuous at $x=2$.
To say that the limit is $4$ means that $f(x) = \dfrac{x^2-4}{x-2}$ can be made as close to $4$ as desired by making $x$ close enough, but not equal, to $2$.
Suppose you want to make $f(x)$ between $4-0.0000001$ and $4+0.0000001$.  There is some number $\delta$ so small that if $x$ is between $2-\delta$ and $2+\delta$ but not exactly $2$, then $f(x)$ is between those bounds.  And if you want $f(x)$ to be between $4-0.0000000000001$ and $4+0.0000000000001$, you can guarantee that by making $\delta$ smaller.
