rudransh verma said:
But I don’t think it will ever stop at a fixed value.It will come as 4.000000000000000000000000000000000000….and then some digit….
This is not a correct intuition for how limits work.
However, you are correct that it never stops at a fixed value. That is also not how limits work.
So let us work on that. What
does it mean when we talk about the limit of a function at a point? In particular, let us use the example at hand. What is the limit of ##f(x) = \frac{(4 + 4 \Delta x + (\Delta x)^2) - 4}{\Delta x}## as ##\Delta x## approaches zero?
You do agree, I hope, that the above limit matches the definition of the derivative of the function ##x^2## at the point 2. We can do a little simplification in the numerator of course so that our starting point is:
$$\lim_{\Delta x \to 0}\frac{4\Delta x + (\Delta x)^2}{\Delta x}$$If we were listening to the teacher, blindly accepting the rules of limits that we are taught, nodding our heads and saying "yes ma'am" we could glance at this limit and immediately say that it is equal to four. But that is not good enough. We want to know why it is exactly 4 and not something like 4.000<infinitely many zeroes>1.
The limit of a function at a point is
not the value of the function somewhere
very near that point.
The limit of a function at a point is
not necessarily the value of the function
at the point. [For a
continuous function, the value of the function at a point will be equal to the limit of the function at that point].
The limit of a function at a point
is the
value (if there is one) that is approached ever more closely as one evaluates the function every more closely to the point.
The standard way of expressing this involves epsilons and deltas. One can quickly google up many copies of the definition. For example:
https://brilliant.org/wiki/epsilon-delta-definition-of-a-limit/ said:
Limit of a function (##\varepsilon-\delta## definition)
Let ##f(x)## be a function defined on an open interval around ##x_0## (##f(x_0)## need not be defined). We say that the limit of ##f(x)## as ##x## approaches ##x_0## is ##L##, i.e.
$$\lim _{ x \to x_{0} }{f(x) } = L$$
if for every ##\varepsilon > 0## there exists ##\delta >0## such that for all ##x##:$$0 < \left| x - x_{0} \right |<\delta \textrm{ } \implies \textrm{ } \left |f(x) - L \right| < \varepsilon$$
[Took a lot of work restoring the embedded ##\LaTeX## in that quote]
This epsilon-delta formulation can be difficult to grasp. Its advantage is that it takes the idea of time and "approaching" out of the picture. The idea of getting closer over time is OK for an intuition, but it is not OK for formal mathematics.
The idea is almost that of a game. You decide how close you want to the formula to get to the limit of 4. That is, you pick an epsilon (##\varepsilon##). I then tell you how close you need to stay to ##\Delta x## = 0. That is, I pick a delta (##\delta##).
If I can always successfully pick a delta (##\delta##) so that value of ##\frac{4\Delta x + \Delta x^2}{\Delta x}## stays within epsilon (##\varepsilon##) of 4 whenever ##\Delta x## is within delta (##\delta##) of zero then the I win and the limit is 4. [We never consider the case where ##\Delta x## is exactly zero]
If you can successfully pick an epsilon (##\varepsilon##) so that I cannot succeed then you win and the limit is not 4 (and may not even exist).
The derivative of x^2 at 2 is exactly 4. It is not an approximation. It is exact.