difference between tending to zero and close to zero In calculus we are taught that the derivative of function $y$ with respect to $x$ is defined as "the quantity which $\dfrac{\bigtriangleup{y}}{\bigtriangleup{x}}$ tends to when $\bigtriangleup{x}$ tends to zero"
Can we also define it as "the ratio of $\dfrac{\bigtriangleup{y}}{\bigtriangleup{x}}$ when $\bigtriangleup{x}$ is a quantity very close to zero"?
 A: Let $y = x^2$.  Then $\dfrac{dy}{dx} = 2x$ for the usual reasons
$\lim_{\Delta x \rightarrow 0} \dfrac {\Delta y}{\Delta x} = \lim \dfrac {[x^2 + 2x\Delta x + (\Delta x)^2] -[x^2]}{\Delta x} = \lim (2x + \Delta x)= 2x$.
But if we chose to say that the derivative is a value where $\Delta x$ is a number "very near zero" (I'm going to ignore the difficulty of how one would define such an ambiguous expression) so that $\Delta x = \delta >0$ but $\delta$ is "small".
Then $\dfrac {\Delta y}{\Delta x} = \dfrac {(x+\delta)^2 - x^2}{\delta}  \dfrac {x^2 + 2x\delta + \delta^2 }{\delta} = 2x + \delta  \ne 2x$.
So, no, that is a different answer.  You can't say $\Delta x$ isn't zero at the beginning and then say "well $\Delta x$ very close to $0$ so we can ignore it" in the end.
A: You can consider the derivative to be the shadow of the ratio $\frac{\Delta y}{\Delta x}$ when $\Delta x$ is infinitesimal.  Alternatively, you can exploit the kind of relation Leibniz had in mind, which was a relation of infinite proximity. Leibniz wrote in his articles that when he speaks of equality, he does not mean strict equality but rather equality up to a negligible term.
Leibniz did not distinguish in notation between strict equality and equality up to a neglibigle infinitesimal, but he did use an alternative notation ${}_{\ulcorner\!\urcorner}\,$ for such a relation so it may be instructive to express this thought as follows.  If $y=f(x)$ then one can write that $f'(x)\;{}_{\ulcorner\!\urcorner}\,\frac{\Delta y}{\Delta x}$ while understanding that $f'(x)$ has to be real (or, as Leibniz would have put it, "assignable").
Shadow is the same as the standard part.
Incidentally, Leibniz's remarks about a notion of equality "up to a negligible term" show that Bishop Berkeley's criticism of the calculus had no merit; see this article for the details.
A: Tending to zero just means, with the "change in X" you can go as close as to zero as you want, correspondingly the (change in Y/ change in X) value will go closer and closer to some number. But if you substitute the value zero or any other value close to zero, you might not end up with that same number towards which we were moving. So the movement is the most important thing here. Its like the function indirectly tells us its value by moving closer and closer to that number. In the book I used, it was mentioned that this value can be seen only after removing the common factor between the numerator and denominator else both would move towards zero.
