Limit of a function

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 x₀ (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 x₀ (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.

History

See mathematical analysis.

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

${\displaystyle \lim _{x\to p}f(x)=L}$ 

if and only if

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

${\displaystyle \lim _{x\to p}f(x)=L}$ 

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

It is just a particular case of functions on metric spaces, with both M and N are the real numbers.

Or we write

${\displaystyle \lim _{x\to p}f(x)=\infty }$ 

if and only if

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;

or we write

${\displaystyle \lim _{x\to p}f(x)=-\infty }$ 

if and only if

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;

If, in the definitions, x-p is used instead of |x-p|, then we get a right-handed limit, denoted by lim_x→p+. If p-x is used, we get a left-handed limit, denoted by lim_x→p-.

Limit of function at infinity

Suppose f(x) is a real-valued function. We can also consider the limit of function when x increases or decreases indefinitely.

We write

${\displaystyle \lim _{x\to \infty }f(x)=L}$ 

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

or we write

${\displaystyle \lim _{x\to \infty }f(x)=\infty }$ 

for every R > 0 there exists S >0 such that for all real numbers x>S, we have f(x)>R;.

Similarly, we can define ${\displaystyle \lim _{x\to \infty }f(x)=-\infty ,\lim _{x\to -\infty }f(x)=L,\lim _{x\to -\infty }f(x)=\infty ,\lim _{x\to -\infty }f(x)=-\infty }$ .

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

If the limit at infinity exists, it represents a horizontal asymptote at x = L. Polynomials do not have horizontal asymptotes; they may occur with rational functions.

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

${\displaystyle \lim _{x\to p}f(x)=L}$ 

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

It is just a particular case of functions over metric spaces with both M and N are the complex plane.

Limit of a function at infinity

We write

${\displaystyle \lim _{x\to \infty }f(x)=L}$ 

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

${\displaystyle \lim _{x\to 3}x^{2}=9}$	The limit of x2 as x approaches 3 is 9. In this case, the function happens to be continuous and the value is well-defined at the point, so the limit is equal to the direct evaluation of the function.
${\displaystyle \lim _{x\to 0}x^{x}=1}$	The limit of xx as x approaches 0 is 1.
${\displaystyle \lim _{x\to 0}{1 \over x}={\mbox{Undefined}}}$  ${\displaystyle \lim _{x\to 0^{+}}{1 \over x}=+\infty }$	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 +∞.
${\displaystyle \lim _{x\to 0^{+}}{|x| \over x}=1}$  ${\displaystyle \lim _{x\to 0^{-}}{|x| \over x}=-1}$	The one-sided limit of |x|/x as x approaches 0 is 1 from the positive side and -1 from the negative side.
${\displaystyle \lim _{x\to \infty }x\sin {1 \over x}=1}$	The limit of x sin(1/x) as x approaches positive infinity is 1.
${\displaystyle \lim _{|x|\to \infty }x^{-a}=0{\mbox{ if }}a\in \mathbb {R} ;a>0;x\in \mathbb {C} }$	Any negative power function approaches 0 as the magnitude of x approaches infinity.
${\displaystyle \lim _{x\to \infty }{x^{a} \over b^{x}}=0{\mbox{ if }}a,b\in \mathbb {R} ;b>0}$	Any power function vanishes in magnitude compared to any increasing exponential function as x approaches infinity.
${\displaystyle \lim _{x\to \infty }{\log _{b}x \over x^{a}}=0{\mbox{ if }}a,b\in \mathbb {R} ;a>0;b>0}$	Any logarithm function vanishes in magnitude compared to any positive power function as x approaches infinity.
${\displaystyle \lim _{x\to \infty }{a^{x} \over x!}=0{\mbox{ if }}a\in \mathbb {R} }$	Any exponential function vanishes in magnitude compared to any factorial function as x approaches infinity.

Functions on metric spaces

• 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.

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.

In the case that f is real-valued, then it is also equivalent to saying that both the right-handed limit or left-handed limit of f at p are 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:

${\displaystyle {\begin{matrix}\lim _{x\to p}&f(x)+g(x)&=&\lim _{x\to p}f(x)+\lim _{x\to p}g(x)\\\lim _{x\to p}&f(x)-g(x)&=&\lim _{x\to p}f(x)-\lim _{x\to p}g(x)\\\lim _{x\to p}&f(x)g(x)&=&\lim _{x\to p}f(x)\cdot \lim _{x\to p}g(x)\\\lim _{x\to p}&{f(x) \over g(x)}&=&{\lim _{x\to p}f(x) \over \lim _{x\to p}g(x)}\end{matrix}}}$ 

(the latter provided that the denominator is non-zero).

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. Indeterminate 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)