Nehtdx

Limits tell us how functions behave at $xto a$, not how they behave at $x = a$. However, in limits we plug $x = a$ as an approximation of $xto a$, so: why the limits give us the exact value of slope of the tangent line, despite being just an approximation of what happend around $x$?

The idea is that the approximation gets better and better as $x$ gets closer to $a$, and, so to say, at $x=a$ it is no more approximative.

Below, a function in blue, the tangent in green and the difference in magenta. As you see, the difference decreases very quickly close to the tangency point.

This is not achieved with a different slope.

A line by line correction of OP that should answer the confusions therein:

(1) Limits tell us how a function behaves nearby $x=a.$ The function doesn't even have to be defined at $a.$ If it is also defined there, then the limit tells us in addition what's happening at $a,$ contrary to your first sentence. But the important thing to note is that limits tell us how a function is behaving in the neighborhood of some point in its domain.

(2) From above, it follows that $f(a)$ need not exist at all in order for $lim_xto af(x)$ to exist. Indeed, even if $f(a)$ exists, it need not coincide with the limiting value. In essence, limiting values have nothing essentially to do with values. But in cases where $lim_xto af(x)$ exists and coincides with $f(a),$ we may use this fact to evaluate the limiting value, but it is no way an approximation. There's nothing about approximations here. The conditions are all equalities. Again, the salient point here is that we can't always evaluate limits, when they exist, by a substitution as you think. This only applies to functions continuous at the point in question.

(3) You have yet to understand the concept of limit properly. It is not an approximation to anything, but rather that towards which we approximate; thus by definition it will sometimes transcend every approximant. Rather, limits are a way to extend usual concepts into wider domains of applicability in such a way that the new objects properly generalise the old -- that is, they possess all important properties of the old objects, and contain the latter as a subclass. In particular here, bringing in limiting operations helps us to assign a definite meaning to some sequences of approximations that distinguish themselves from others. Note that whereas some terms of these types of sequence approximate the limiting value, this value itself is decidedly not an approximation, but a uniquely defined value, fixed by this sequence (and an assumption about nonempty subsets of real numbers that genuinely and smoothly extends the concept of maximum or minimum values of nonempty subsets of numbers). This well-defined limiting value, whenever it exists, has been shown many times over to legitimately (in a sense that can be made precise) extend the domain of application of the concept under study. Here, it follows that whenever a sequence of approximating slopes converges, we can choose to call (by a justified extension) the limit a slope too, indeed the slope, defined by that sequence. That's simply what it means to work with real numbers. If you've not understood this property (technically called completeness) then you've not begun to understand the real numbers.

Well, if you have two points $(x_1,y_1)$ and $(x_2,y_2)$ in the plane, then the slope of the unique line through these points is
$$ m = fracy_2-y_1x_2-x_1.$$
If you consider a (differentiable) function $f:Bbb RrightarrowBbb R$, then the slope of the line between $(x_1,f(x_1))$ and $(x_2,f(x_2))$ is
$$ m = fracf(x_2)-f(x_1)x_2-x_1.$$
If you take the points $x_0$ and $x$ at the x-axis, then consider the quotient
$$fracf(x)-f(x_0)x-x_0$$
when $x$ moves to $x_0$: $xrightarrow x_0$.
Then this quotient becomes (in the limit) the slope of the tangent line of $f$ at the point $x_0$. This is particularly easy to understand if you draw a picture.

Denote the slope in the limit as $f'(x_0)$. Then the above quotient gives
$$f'(x_0) = fracf(x)-f(x_0)x-x_0$$
and so the equation of the tangent line
$$f(x) = f'(x_0)(x-x_0) + f(x_0).$$

I think the problem with your thinking is that, you are assuming that there is something that is predefined by "god" which is the slope at a point that has exact value, and we are trying to reach it by the processes of taking a limit, and we can never reach that value. But that is not the case. What we do, is that we define the notion of the slope at a point , by the processes of taking a limit. And what ever the exact value that comes out of the limit, is what we call the slope at that point.

I guess that what is bothering you, is the notion of a limit that you learn at calculus , and i felt the same when i first learn it, because at some stage of the problem we say that x approaches some value a , then at another stage we substitute a for x. But to understand what is actually going on, you have to understand the epsilon-delta definition of a limit.
see:https://en.wikipedia.org/wiki/(%CE%B5,_%CE%B4)-definition_of_limit