basic concepts of limits

Question

Suppose I have a function:

$ f(x)=x$

Now I want to calculate the limits: $\lim\limits_{x\rightarrow1}f(x)=f(1)$

As wiki_limits said,

the limit of $f$ of $x$, as $x$ approaches 1, is $f(1)$.

So my question here is

1 Does it mean the limit calculation are actually is a approximated kind of calculation? It is not normal precise calculation such as $1+1=2$

2 If all limit calculation are approximated results, does it mean that Caculus are about approximated calculation? (Based on my current knowledge, just begin to learn single calculus)

Answer 1

From interaction with OP via comments it appears that OP is of the view that limits are used to deal with approximate calculation. Thus he mentions via comment: "Does $x \to 1$ mean $x = 1$? It seems to me that $x$ will be as much near as possible to $1$ but will not be equal to 1. So the approximation comes here."

I gave a reply via comment but then felt like adding some explanation and hence this answer came into existence.

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Perhaps it is a common misconception that the meaning of the symbol $x \to 1$ (as opposed to $x = 1$) is that $x$ takes values near and near to $1$ so that some sort of endless activity of assigning values (near and near to $1$) to $x$ is ongoing here.

This in reality is a deep misconception and it is probably the result of the efforts of various textbook authors and classroom instructors to explain the notion of limits in intuitive manner. Sometimes such intuitive explanations cross the line and deviate too much from the concept being explained. The symbol $x \to a$ does not have an independent meaning in isolation, but rather it is given a meaning in a proper context.

One such context is the phrase "$f(x) \to L$ as $x \to a$" or equivalently the notation $\lim\limits_{x \to a}f(x) = L$. In such a context the symbol $x \to a$ does not have anything to do with assigning any set of values to $x$. Rather this phrase is guaranteeing the truth of an infinite number of logical statements. This is something difficult to grasp initially and hence some symbolism/formalism may be avoided while explaining this concept. Informally, the phrase $f(x) \to L$ as $x \to a$ means that it is possible to ensure that all the values of $f(x)$ lie as near to $L$ as we please by choosing the values of $x$ sufficiently close to but not equal to $a$. Thus we can think of "ensuring $f(x)$ as close to $L$ as we please" as a goal (in reality it corresponds to an infinite number of goals because of the use of phrase "as close ... as we please") and choosing values of $x$ close to but not equal to $a$ as a means to ensure that this goal can perhaps be achieved. The statement $\lim\limits_{x \to a}f(x) = L$ tells us that this complicated sort of goal is possible to achieve in the manner indicated.

To put it more crudely: "$f(x) \to L$ as $x \to a$" does not mean that $x$ is taking values near $a$ and as a result of it values of $f(x)$ are near $L$. Rather it means that if $x$ takes values too close to $a$ then values of $f(x)$ will be near $L$. Note that there is an implication involved here because of "if". We don't actually care/know/state what the value of $x$ is but we care/know/state that if values of $x$ are near $a$ then values of $f$ are near $L$. This crude version seems to imply that the part dealing with hypotheses "if $x$ is near $a$" is more important/fundamental than the part dealing with conclusion "values of $f$ are near $L$" whereas as in reality it is the reverse and the focus of limit concept is to ensure that the values of $f$ can be constrained in a very specific manner by constraining the values of $x$ in another specific manner.

Thus the concept of limit does not involve assigning a set of values to the independent variable $x$ and thinking about the values of the dependent variable / function $f(x)$. Rather it involves asserting that the values of $f$ can be ensured to behave in a certain manner (or can be ensured to have a specific trend) by restricting the values of $x$ in another specific manner.

Also if one asks the question "if $x \to a$ then what can you say about the value of $x$?" then the right answer is not that the values of $x$ are near $a$, but rather the right answer is "the symbol $x \to a$ alone cannot be interpreted to deduce any information about values of $x$ except the bare minimum assurance that $x \neq a$."

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Regarding approximations it is wrong to think that limits are a tool to calculate the approximate value of $f(x)$ by choosing an approximate value of $x$ near $a$. Limits are well defined operations on functions and they are not used to approximate the value of a function, but are rather used to study the behavior/trend of values of a function in a specific manner.

On the other hand there are many approximation techniques which have their basis in calculus. A famous example is the Newton Raphson method to approximately find roots of equations (both polynomial and non-polynomial type). Using this method we get the famous technique of approximating square roots. If we wish to find an approximation to $\sqrt{A}$ then we start with any positive number $a$ and calculate the numbers $f(a), f(f(a)), f(f(f(a)))$ and so on where $$f(x) = \dfrac{1}{2}\left(x + \dfrac{A}{x}\right)$$ Each of the values $a, f(a), f(f(a)), \ldots$ is a better approximation to $\sqrt{A}$ than the previous one in sequence. You should check this method with $A = 2, a = 1$. Another example is the use of Taylor series to evaluate values of corresponding functions.

Answer 2

The notion of limit is, most generally, a topological notion. It is defined in terms of the topology of the space in question. This of course begs the question: what is a topology? Formally, a topology, $\tau$, for a set $X$ is a collection of subsets of that set, defined to be open, which satisfies the following properties:

1: The empty set $\{\}$ and the entire set, $X$ are elements of the topology, $\tau$.

2: Given any collection of sets in the topology, $U_i\in\tau$ for $i\in I$ ($I$ is an index set), the union of those sets, $\bigcup_{i\in I} U_i$ is also in the topology, $\bigcup_{i\in I} U_i\in\tau$.

3: Given any finite collection of sets in the topology, $U_i\in\tau$ for $i\in I$, the intersection of those sets, $\bigcap_{i\in I} U_i$ is also in the topology, $\bigcap_{i\in I} U_i\in\tau$.

The point of the above abstract definition is that the resulting structure gives a notion of nearness, via the open sets defined by the given topology, that is less rigid than distance (to fully appreciate what is meant by that, you would need to study topology a fair bit).

With topologies defined, one can then define the notion of a limit of a function rigorously:

A limit of a function, $f$ between topological spaces, $(X,\tau)$ and $(Y,\sigma)$, as $x\rightarrow x_0$ is a point $y_0\in Y$ such that:

Given any open set $V\in\sigma$ containing $y_0$, there exists an open set in $U$ containing $x_0$ such that the image under $f$, of $U-\{ x_0\}$ ($U$ with the point $x_0$ removed from it), $f(U-\{ x_0\})$ is contained in $V$, $f(U-\{ x_0\})\subseteq V$.

Two things to note: First of all, existence of such a point $y_0$ is not guaranteed, and indeed there are many examples of limits that do not exist. Second, there is no immediate guarantee that the $y_0$ is unique, there may be many such points, hence in general one can only speak of a limit rather than the limit. However, most topologies of interest outside of the study of topology itself satisfy a property called the Hausdorff property, which guarantees that when a limit exists, it is unique, hence in Hausdorff topological spaces we can speak of the limit.

The point of the definition of limit above, is that the function, $f$ can be said to eventually be within every neighborhood of the limit point (the term neighborhood is a synonym for open set here).

Bringing the above abstraction back down to the case of interest: The real numbers has a natural choice of topology, which is defined via a basis:

A basis for a topology is a subset of that topology, from which one can reconstruct the rest of the topology via unions. For the real numbers, this basis is the open intervals: $(a,b)$ (with $a<b$). Hence an arbitrary element of the standard topology for the reals looks like a union of open intervals:

$\bigcup_{i\in I}(a_i,b_i)$, where $i\in I$

If consider a limit point $y_0$, provided that it exists, we speak of the open neighborhoods around that point, those being any open interval $(a,b)$ such that $y_0\in (a,b)$ (more generally given an element of the standard topology of the reals, $y_0$ will lie in at least one of the open intervals in the union, hence we can concentrate on just basis elements). The definition of limit in this context becomes: given any open interval $(a,b)$ containing $y_0$, there exists an open interval $(c,d)$ containing $x_0$ such that the image of $(c,d)-\{ x_0\} =(c,x_0)\cup (x_0,d)$, $f((c,x_0)\cup (x_0,d))$ is contained in the other open interval $f((c,x_0)\cup (x_0,d))\subseteq (a,b)$. The formal $\epsilon$-$\delta$ definition of a limit that you will encounter in most formal introductions to limits (e.g. in most rigorous calculus texts) is nothing more than a restatement of the above without specific reference to the topology involved (since much of the motivation for developing topology came from attempts to extend the tools and techniques of calculus to more general mathematical contexts, and, historically, calculus pre-dates topology by roughly 3 centuries, and even limits pre-date topology by at least 50 years, this is pedagogically justified).

The intuitive way to think about limits in the context at hand is to ask: What is the function doing around the $x=x_0$. Is it always staying around some particular value, $y_0$? If yes, then, intuitively, it has a limit at $x=x_0$ of $y_0$. The above makes that intuition precise in a mathematically rigorous way.

Answer 3

Yes to both questions, except that in a limit your approximation is so close to the actual value, you can't tell the difference. And, in some cases, the 'actual value' does not exist but the limit does. So I'd describe these kind of limits as being close to a value that 'would be there' if it were defined in the first place. (Look up removable discontinuity for example.)

This is all made rigorous by epsilon and deltas, which in Newton's and Leibniz's Calculus were called things like fluxions and infinitesimals. The more you study Calculus and limits in general, the more clear these definitions and ideas will be for you...