In mathematics, the limit of a function is a fundamental concept in mathematical analysis.
Rather informally, to say that a function f has a limit y when x tends to a value x_{0} (or to the infinity), is to say that the values taken by the expression f(x) get close to y when x gets close to x_{0} (or gets infinitely big). Formal definitions, first devised around the end of the 19th century, are given below.
See net (topology) for a generalisation of the concept of limit.
Table of contents |
2 Formal definition 3 Examples 4 Properties 5 See also 6 References |
History
Formal definition
Functions on metric spaces
Suppose f : (M,d_{M}) -> (N,d_{N}) is a map between two metric spaces, p∈M and L∈N. We say that "the limit of f(x) is L as x approaches p" and write
- for every ε > 0 there exists a δ > 0 such that for all x in M with 0 < d_{M}(x, p) < δ, we have d_{N}(f(x), L) < ε.
Real-valued functions
The real number line is itself a metric space. But it has some different types of limits.
Limit of a function at a point
Suppose f is a real-valued function, then we write
if and only if- for every ε > 0 there exists a δ >0 such that for all real numbers x with 0<|x-p|<δ, we have |f(x)-L|<ε
Or we write
- for every R > 0 there exists a δ >0 such that for all real numbers x with 0<|x-p|<δ, we have f(x)>R;
- for every R < 0 there exists a δ >0 such that for all real numbers x with 0<|x-p|<δ, we have f(x)<R;
Limit of function at infinity
if and only if- for every ε > 0 there exists S >0 such that for all real numbers x>S, we have |f(x)-L|<ε
- for every R > 0 there exists S >0 such that for all real numbers x>S, we have f(x)>R;.
There are three basic rules for evaluating limits at infinity for a rational function f(x) = p(x)/q(x):
- If the degree of p is greater than the degree of q, then the limit is positive or negative infinity depending on the signs of the leading coefficients
- If the degree of p and q are equal, the limit is the leading coefficient of p divided by the leading coefficient of q
- If the degree of p is less than the degree of q, the limit is 0
Complex-valued functions
The complex plane is also a metric space. There are two different types of limits when we consider complex-valued functions.
Limit of a function at a point
Suppose f is a complex-valued function, then we write
if and only if- for every ε > 0 there exists a δ >0 such that for all real numbers x with 0<|x-p|<δ, we have |f(x)-L|<ε
Limit of a function at infinity
if and only if- for every ε > 0 there exists S >0 such that for all complex numbers |x|>S, we have |f(x)-L|<ε
Examples
Real-valued functions
- 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 x^{2} 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 x^{x} as x approaches 0 is 1.
- The limit of ((a + x)^{2} - a^{2} ) / x as x approaches 0 is 2a.
- The one-sided limit of sqrt(x^{2})/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.
Functions on metric spaces
- If z is a complex number with |z| < 1, then the sequence z, z^{2}, z^{3}, ... 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.
Properties
To say that the limit of a function f at p is L is equivalent to saying
- for every convergent sequence (x_{n}) in M - {p} with limit equal to p, the sequence (f(x_{n})) 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 limit of af(x) as x approaches p is aL.
Taking the limit of functions is compatible with the algebraic operations: If
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 ≠ ± ∞
Note that there is no general rule for the case q / 0; it all depends on the way 0 is approached. Interdeterminate forms, 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.
See also
- How to evaluate the limit of a real-valued function
- Limit of a sequence
- Net (topology)
- Big O notation
References
- Visual Calculus by Lawrence S. Husch, University of Tennessee (2001)