What is the mathematical definition of a "limit"?

Answer 1

In mathematics, the concept of a "limit" is used to describe the behavior of a function as its argument either "gets close" to some point, or as it becomes arbitrarily large; or the behavior of a sequence's elements as their index increases indefinitely. Limits are used in calculus and other branches of mathematical analysis to define derivatives and continuity.

The concept of the "limit of a function" is further generalized to the concept of topological net, while the limit of a sequence is closely related to limit and direct limit in category theory.

Main article: Limit of a function
Suppose ƒ(x) is a real function and c is a real number. The expression:


means that ƒ(x) can be made to be as close to L as desired by making x sufficiently close to c. In that case, we say that "the limit of ƒ of x, as x approaches c, is L". Note that this statement can be true even if . Indeed, the function ƒ(x) need not even be defined at c. Two examples help illustrate this.

Consider as x approaches 2. In this case, f(x) is defined at 2 and equals its limit of 0.4:

f(1.9) f(1.99) f(1.999) f(2) f(2.001) f(2.01) f(2.1)
0.4121 0.4012 0.4001 0.4 0.3998 0.3988 0.3882

As x approaches 2, ƒ(x) approaches 0.4 and hence we have . In the case where , ƒ is said to be continuous at x = c. But it is not always the case. Consider


The limit of g(x) as x approaches 2 is 0.4 (just as in ƒ(x)), but ; g is not continuous at x = 2.

Or, consider the case where ƒ(x) is undefined at x = c.


In this case, as x approaches 1, f(x) is undefined at x = 1 but the limit equals 2:

f(0.9) f(0.99) f(0.999) f(1.0) f(1.001) f(1.01) f(1.1)
1.95 1.99 1.999 undef 2.001 2.010 2.10

Thus, x can get as close to 1, so long as it is not equal to 1, so that the limit of f(x) is 2.


Formal definition
Karl Weierstrass formally defined a limit as follows:

Let be a function defined on an open interval containing (except possibly at ) and let be a real number.


means that

for each real there exists a real such that for all x with , we have
The formal definition of a limit is sometimes called the epsilon-delta form because it uses the Greek letters delta (δ) and epsilon (ε). The use of the particular Greek letters δ and ε is merely traditional; the definition would, of course, be unchanged if different letters or symbols were used.

Caution: It should be noted that this definition provides a way to recognize a limit without providing a way to calculate it. One often needs to find a limit using informal methods, especially when f(x) is discontinuous at c, for example, when f is a ratio with a denominator that becomes 0 at c. One should check that the result actually meets the Weierstrass definition in such cases.





Limit of a function at infinity
A related concept to limits as x approaches some finite number is the limit as x approaches positive or negative infinity. This does not literally mean that the difference between x and infinity becomes small, since infinity is not a real number; rather, it means that x either grows without bound positively (positive infinity) or grows without bound negatively (negative infinity).

For example, consider .

f(100) = 1.9802
f(1000) = 1.9980
f(10000) = 1.9998
As x becomes extremely large, the value of f(x) approaches 2, and the value of f(x) can be made as close to 2 as one could wish just by picking x sufficiently large. In this case, we say that the limit of f(x) as x approaches infinity is 2. In mathematical notation,


Formally, we have the definition

if and only if for each ε > 0 there exists an n such that | f(x) − c | < ε whenever x > n.
Note that the n in the definition will generally depend on ε. A similar definition applies for .

If one considers the domain of to be the extended real number line, then the limit of a function at infinity can be considered as a special case of limit of a function at a point.


Limit of a sequence
Main article: Limit of a sequence
Consider the following sequence: 1.79, 1.799, 1.7999,... We could observe that the numbers are "approaching" 1.8, the limit of the sequence.

Formally, suppose x1, x2, ... is a sequence of real numbers. We say that the real number L is the limit of this sequence and we write


if and only if

for every real number ε > 0 there exists a natural number n0 (which will depend on ε) such that for all n > n0 we have | xn − L | < ε.
Intuitively, this means that eventually all elements of the sequence get as close as we want to the limit, since the absolute value | xn − L | can be interpreted as the "distance" between xn and L. Not every sequence has a limit; if it does, we call it convergent, otherwise divergent. One can show that a convergent sequence has only one limit.

The limit of a sequence and the limit of a function are closely related. On one hand, the limit of a sequence is simply the limit at infinity of a function defined on natural numbers. On the other hand, a limit of a function f at x, if it exists, is the same as the limit of the sequence xn = f(x + 1 / n).

Answer 2

I agree with some others here that to a mathematical fact is a truth derived from assumptions and definitions. In a way, the only mathematical facts and theorems are in the form if X_1 and X_2 and X_3 and ... then Y and their proofs are a sequence of statements linked together only by the assumptions X_i. Usually in mathematics, the assumptions include at least the axioms and rules for logic and set theory but not always. I suppose the simplest mathematical fact is, then, the one with the fewest assumptions and fewest steps in the proof. This is likely to be something quite trivial like If X then X which is usually taken as an axiom of logic but the system I'm using has no axioms*, as such, just an idea of the relationship between statements. The proof is also trivial since the sequence of statements linked by assumptions and ending with the conclusion is X. *Other than the definitions of proof and so on above, I may have to presume something about substitutions to make the assumptions work together in a fuller setting but that would be unnecessarily complicated here since I only have one statement. Note: I suppose by "mathematical" you might be willing to work in any conventional axiomatic system and so take some assumptions for free. In this case, I think the simplest proofs are existence proofs and identity proofs. The existence of the empty set using a subset of ZF set theory axioms comes to mind first.

Answer 3

Strictly speaking (and you do need to be precise when it comes to defining concepts in mathematics), the formal definiton can be worded like this:

The function f(x) has a limit L if, for any ε > 0,
you can find a value N such that for all x >= N,
| f(N) - L | < ε.

In other words, the function has a limit L if for any positive value (no matter how infinitely small), you can always find a point where all higher values of the function will be within that range of L.

Answer 4

In mathematics, the concept of a "limit" is used to describe the behavior of a function as its argument either "gets close" to some point, or as it becomes arbitrarily large; or the behavior of a sequence's elements as their index increases indefinitely. Limits are used in calculus and other branches of mathematical analysis to define derivatives and continuity. And if asking another definition of a limit are on the second link.

Answer 5

It depends on which object you're working on (functions or sequences). For real function f: A \to R, where A is an open subset of R. One says that a real number L is the limit of f as x tends to a \in A if the following holds:
for every positive real number e there is a positive real number d (depending on e and a) such that for each x in A that satisfies
|x - a| < d one has | f(x) - L | < e.
The definition is slightly different when L is not a real number, i.e. L is infinity.
Then one says that the limit of f as x tends to a equals infinity (resp. - infinity) if for every positive real number e there is a real number d such that for each x in A that satisfies | x - a | < d one has f(x) > e (resp. f(x) < - e).
You can get more details on wikipedia by typping limit.

Answer 6

A function f(x) is said to approach a limit L as x approaches a if the difference between f(x) and L is numerically less than an arbitrarily small positive number for all values of x in the range of definition that are sufficiently close to a and for which x is not equal a.
We say lim f(x) = L
x --> a, or
f(x) --> L as x --> a

Important rules of limits are:
lim(u+v) = lim u +lim v
lim(u*v) = lim u * lim v
Lim(u/v) = (lim u)/(lim v) lim v not = 0
lim u^n) = (lim u)^n
lim (nth root u) = nth root(lim u)

Answer 7

A limit is the intended height of a function.

Answer 8

a limit is a number the function seems to be approaching at a certain time, x = c.

Answer 9

The value of the function a long way from the origin, what the function is converging to.

Answer 10

The value when it is in some number (related to a function).