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Limit (mathematics)

The mathematical concept of limit is used to describe the behavior of a function as its argument get "close" to some point (or attempts to get close to infinity), or the behavior of the elements of a sequence as their index approaches infinity. Limits are used in calculus (and other branches of mathematical analysis) to define derivatives and continuity.

Table of contents

Informal Introduction

Suppose f(x) is a real function and c is a real number. If the values of f(x) approach (get close to, but don't necessarily reach) the number L, as x approaches c, one can state that "the limit of f(x), as x approaches c, is L" and write

<math>
\lim_{x \to c}f(x) = L. </math>

If the values of f(x) approach infinity as x approaches c, one writes

<math>
\lim_{x \to c}f(x) = \infty. </math> Note, infinity is not a "limit", limits must be real numbers.

Finding limits

In "well-behaved" functions (i.e. in continuous ones), the limit, as x approaches c, can be found by directly substituting c for x. For example, if f(x) = 7; as x approaches 32, the limit is 7 (the limit of a constant is a constant). Another example is f(x) = 2x - 5; in that situation, as x approaches 3, f(x) approaches f(3) = 2·3 - 5 = 1.

Limits are more interesting when they are unreachable; for example: if f(x) = (x³ - 1) / (x - 1) then, x cannot equal 1 (as that would result in division by zero); however, f(x) does approach some number c (as x approaches 1). f(0.9) = 2.71, f(0.99) = 2.9701, f(0.999) = 2.997001, f(1.1) = 3.31, f(1.01) = 3.0301, f(1.001) = 3.003001 -- We see that, as x approaches 1, f(x) approaches 3; however, x never equals 1 and f(x) never equals 3. The limit can be verified using algebra; since: (x2 + x + 1)(x - 1) = (x3 - 1)...if g(x) = x2 + x + 1, g(x) = f(x) (for any x ≠ 1). g(x) and f(x) are identical at every point, except 1; f(x) has a hole at 1. Via direct substitution (x = 1) one can determine the limit of g(x) to be 12 + 1 + 1 = 3; which is the value f(x) approaches, as it approaches the hole.

Here we have used the following rule: if it so happens that if g(x) = f(x) for all values of x, except for c; then, so long as g(x) has a limit, that limit will be equal to the limit of f(x).

The above also shows that whether or not f(c) exists has no bearing on whether or not the limit of f(x) (as x approaches c) exists. If f(c) exists, then its value has also no bearing on the limit.

Limits Do Not Always Exist

Not every function has a limit at every point. Consider:

  • As x approaches 0, f(x) = |x| / x does not approach a limit; the value of f(x) is -1 if x<0 and +1 if x>0. No number L can serve as a limit; we say that the "one-sided limit from the left as x approaches 0" is -1 and the "one-sided limit from the right" is +1.
  • As x approaches 0, f(x) = 1 / x is not approaching a limit; for x>0 it is unbounded above and approaches +∞, for x<0 it is unbounded below and approaches -∞.
  • As x approaches 0, f(x) = sin (1/x) does not approach a limit, as it is oscillates faster and faster.

L'H˘pital's Rule and Division by zero

When trying to evaluation a limit by simply substitution c into the function, one often would have to divide by zero, which is of course impossible. Here l'H˘pital's rule helps: If f(c)=0, g(c)=0, and both derivatives f'(c) and g'(c) are defined, then:

<math>\lim_{x\to c}{f(x)\over g(x)}=\lim_{x\to c}{f'(x)\over g'(x)}</math>

Three Notable Limits

<math>
\lim_{x \to 0}\frac{\sin x}{x} = 1 </math>
<math>
\lim_{x \to 0}\frac{1 - \cos x}{x} = 0 </math>
<math>
\lim_{x \to 0}(1 + x)^{1/x} = e </math>

The "Squeeze Theorem"

If three functions f(x), g(x) and h(x) are given such that h(x) ≤ f(x) ≤ g(x) and if the limit, L, (as x approaches c) of h(x) is equal to the limit (as x approaches c) of g(x); then, the limit of f(x) not only exists, but is also equal to L. The function f(x) is "squeezed" between g(x) and h(x).

Approaching a Limit

One might ask whether there is some relationship between |f(x) - L| (the absolute value [of the function, at x, minus its limit, L, as x approaches c]) and |x - c| (the absolute value [of x minus the number it is approaching]). For every number, ε > 0, there is some number, δ > 0; such that, if 0 < |x - c| < δ; then, |f(x) - L| < ε. In other words, if the distance between x and c is less than δ; then, the distance between f(x) and L is less than ε.

For example; the limit of (3x - 2), as x approaches 3, is 7; f(3) = 7. If one determines that the absolute value [of the function, at x, minus its limit (as x approaches 3)], should be less than 0.003 (that is, one is attempting to determine what value of x will generate an f(x) within 0.003 of the limit of f(x), as x approaches c = 3); then, one can write: |(3x - 2) - 7| = |3x - 9| = 3|x - 3| < 0.003. Noting that 0 < |x - c| < δ and that, in this situation, c = 3; we can write 0 < |x - 3| < δ and since 3|x - 3| < 0.003; it is only logical to conclude that |x - 3| < 0.003 / 3 = 0.001; and thus, any value within 0.001, of 3, will produce a value within 0.003 of f(x)'s limit (as x approaches 3); that is, a value within 0.003 of 7. For instance, f(3.001)=7.003.

Formal definitions

The rest of this article presents the concept of limit in increasing generality, starting with sequences and functions of real numbers, then metric spaces, and culminating in the most general concept, limits of nets on topological spaces. This follows a more and more general definition of how "close to" is evaluated.

Limit of a sequence of real numbers

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

<math>
\lim_{n \to \infty}x_n = L </math>

if and only if

for every 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.

Examples

  • The sequence 1/1, 1/2, 1/3, 1/4, ... of real numbers converges with limit 0.
  • The sequence 1, -1, 1, -1, 1, ... is divergent.
  • The sequence 1/2, 1/2 + 1/4, 1/2 + 1/4 + 1/8, 1/2 + 1/4 + 1/8 + 1/16, ... converges with limit 1. This is an example of an infinite series.
  • If a is a real number with absolute value |a| < 1, then the sequence an has limit 0. If 0 < a ≤ 1, then the sequence a1/n has limit 1.

Properties

A function f : R -> R is continuous if and only if it is compatible with limits in the following sense:

if (xn) is any convergent sequence in R with limit L, then the sequence (f(xn)) converges with limit f(L).

A subsequence of the sequence (xn) is a sequence of the form (xa(n)) where the a(n) are natural numbers with a(n) < a(n+1) for all n. Intuitively, a subsequence omits some elements of the original sequence. A sequence is convergent if and only if all of its subsequences converge towards the same limit.

The limit operation is a linear operator in the following sense: if (xn) and (yn) are convergent sequences of real numbers and lim xn = L and lim yn = P, then the sequence (xn + yn) is also convergent and has limit L + P. If a is a real number, then the sequence (a xn) is convergent with limit aL. Therefore, the set c of all convergent sequences of real numbers is a real vector space and the limit operation is a linear operator from c to the real numbers.

If (xn) and (yn) are convergent sequences of real numbers with limits L and P respectively, then the sequence (xnyn) is convergent with limit LP. If neither P nor any of the yn is zero, then the sequence (xn/yn) is convergent with limit L/P.

Every convergent sequence is a Cauchy sequence and hence bounded. If (xn) is a bounded sequence of real numbers which is increasing (i.e. xnxn+1 for all n), then it is necessarily convergent. More generally, every Cauchy sequence of real numbers has a limit, or short: the real numbers are complete.

A sequence of real numbers is convergent if and only if its limit inferior and limit superior coincide and are both finite.

Limit of a function at a point

Suppose f : U -> R is a function, where U is a subset of the real numbers. If p and L are two real numbers, we say that the limit of f(x) as x approaches p is L and write

<math>
\lim_{x \to p}f(x) = L </math>

if and only if

for every ε > 0 there exists a δ > 0 such that for all x in U with 0 < |x - p| < δ, we have |f(x) - L| < ε.

This is equivalent to saying

for every convergent sequence (xn) in U - {p} with limit equal to p, the sequence (f(xn)) converges with limit L.

Note that the function does not have to be defined at p, and in any case, the value f(p) is irrelevant for the determination of the limit of f at p.

We also consider situations where either p or L or both are positive or negative infinity. We say that f(x) approaches positive infinity (+∞) as x approaches p if and only if

for every R > 0 there exists a δ > 0 such that whenever 0 < |x - p| < δ then f(x) > R.

We say that the limit of f(x) as x approaches positive infinity is L if and only if

for every ε > 0 there exists an S > 0 such that whenever x > S, then |f(x) - L| < ε.

Finally, we say that the limit of f(x) is positive infinity as x approaches positive infinity if and only if

for every R > 0 there exists an S > 0 such that whenever x > R, then f(x) > S.
The definitions for negative infinity are analogous.

Occasionally, it is useful to approach the point p only from one side. The one-sided limit of f(x) as x approaches p from the right is L, written as

<math>
\lim_{x \to p+}f(x) = L </math>

if and only if

for every ε > 0 there exists a δ > 0 such that for all x in U with 0 < x - p < δ, we have |f(x) - L| < ε.

Left-sided limits are obtained by replacing x - p in the last definition by p - x. By replacing ε by S as above, we can also define one-sided limits that are infinite.

Examples

  • The limit of 1/x as x approaches infinity is 0.
  • The two-sided limit of 1/x as x approaches 0 does not exist. The limit of 1/x as x approaches 0 from the right is +∞.
  • The limit of x2 as x approaches 3 of is 9. (In this case the value of the function happens to be well defined at the point, and the function's value is the same as its limit.)
  • The limit of xx as x approaches 0 is 1.
  • The limit of ((a + x)2 - a2 ) / x as x approaches 0 is 2a.
  • The one-sided limit of sqrt(x2)/x as x approaches 0 from the right is 1; the one-sided limit from the left is -1.
  • The limit of x sin(1/x) as x approaches positive infinity is 1.
  • The limit of (cos(x) - 1)/x as x approaches 0 is 0.

Properties

If the limit of f(x) as x approaches p exists (which need not be the case), and if there exists at least one sequence (xn) with elements in U - {p} and limit equal to p, then the limit of f(x) as x approaches p is uniquely determined by f and p.

The two-sided limit of f(x) as x approaches p exists if and only if the left-sided and right-sided limits exist and are equal.

The function f is continuous at the point p if and only if the two-sided limit of f(x) as x approaches p is finite and equal to f(p).

Taking the limit of functions is compatible with the algebraic operations: If

<math>
\lim_{x \to p}f_1 (x) = L_1 </math> and
<math>
\lim_{x \to p}f_2 (x) = L_2 </math>

then

<math>
\lim_{x \to p}(f_1 (x) + f_2 (x)) = L_1 + L_2 </math>

and

<math>
\lim_{x \to p}(f_1 (x) \times f_2 (x)) = L_1 \times L_2 </math>

and

<math>
\lim_{x \to p} \left( \frac{f_1 (x)} {f_2 (x)}\right) = \frac{L_1} {L_2} </math>

(the latter provided that f2(x) is non-zero in a neighborhood of p and L2 is non-zero as well).

These rules are also valid for one-sided limits, for the case p = ±∞, and also for infinite limits using the rules

  • q + ∞ = ∞ for q ≠ -∞
  • q × ∞ = ∞ if q > 0
  • q × ∞ = -∞ if q < 0
  • q / ∞ = 0 if q ≠ ± ∞
(see extended real number line).

Note that there is no general rule for the case q / 0; it all depends on the way 0 is approached. Some cases, for instance 0/0, 0×∞ ∞-∞ or ∞/∞, are also not covered by these rules but the corresponding limits can usually be determined with l'Hôpital's rule.

Metric spaces

The real numbers form a metric space if we use the distance function given by the absolute value: d(x,y) = |x - y|. The same is true for the complex numbers. Furthermore, the Euclidean space Rn forms a metric space with the metric given by the euclidean distance. These three will be our motivating examples for extending the limit definitions given above.

If (xn) is a sequence in the metric space (M, d), we say that the sequence has limit L iff for every ε>0 there exists a natural number n0 such that for all n>n0 we have d(xn, L) < ε.

If the metric space (M, d) is complete (which is true for the real and complex numbers and Euclidean space, and all other Banach spaces), then one can establish the convergence of a sequence in M by showing that it is a Cauchy sequence. The advantage of this approach is that one need not know the limit in advance in order to do this.

If M is a real or complex normed vector space, then the limit operation is linear, as explained above for the case of sequences of real numbers.

Now suppose f : M -> N is a map between two metric spaces, p is an element of M and L is an element of N. We say that the limit of f(x) as x approaches p is q and write

<math>
\lim_{x \to p}f(x) = q </math>

if and only if

for every ε > 0 there exists a δ > 0 such that for all x in M with 0 < d(x, p) < δ, we have d(f(x), L) < ε.

This is equivalent to saying

for every convergent sequence (xn) in M - {p} with limit equal to p, the sequence (f(xn)) converges with limit L.

The function f is continuous at p if and only if the limit of f(x) as x approaches p exists and is equal to f(p). Equivalently, f transforms every sequence in M which converges towards p into a sequence in N which converges towards f(p).

Again, if N is a normed vector space, then the limit operation is linear in the following sense: if the limit of f(x) as x approaches p is L and the limit of g(x) as x approaches p is P, then the limit of f(x) + g(x) as x approaches p is L + P. If a is a scalar from the base field, then the the limit of af(x) as x approaches p is aL.

If N is R, we can define infinite limits; if M is R, we can define one-sided limits in analogy to the definitions given earlier.

Examples

  • If z is a complex number with |z| < 1, then the sequence z, z2, z3, ... of complex numbers converges with limit 0. Geometrically, these numbers "spiral into" the origin, following a logarithmic spiral.
  • In the metric space C[a,b] of all continuous functions defined on the interval [a,b], with distance arising from the supremum norm, every element can be written as the limit of a sequence of polynomial functions. This is the content of the Stone-Weierstrass theorem.

Generalizations

All of the above notions of limit can be unified and generalized to arbitrary topological spaces by introducing nets and defining their limits. The article on nets elaborates on this.

An alternative is the concept of limit for filters on topological spaces.

See also

References



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