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LIMIT OF A FUNCTION

Topics in calculus

Fundamental theorem
Limits of functions
Continuity
Vector calculus
Tensor calculus
Mean value theorem
Differentiation

Product rule
Quotient rule
Chain rule
Implicit differentiation
Taylor's theorem
Related rates
Table of derivatives
Integration

Lists of integrals
Improper integrals
Integration by: parts, disks,
cylindrical shells, substitution,
trigonometric substitution

In mathematics, the limit of a function is a fundamental concept in mathematical analysis. Informally, a function f has limit L at a point p when we can make the value of f as close to L as we want, by taking points close enough to p. Formal definitions, first devised around the end of the 19th century, are given below.

Contents

History

See mathematical analysis.

Definition

To motivate the definition of a limit, consider the following informal statement:

A real-valued function f(x) has limit L as x approaches the real number c if the values of f(x) become closer and closer to L as the value of x gets closer and closer to c.

To contextualize this informal statement, imagine a traveler walking along the graph of y=f(x). Her horizontal position is measured by the value of x: this is like the position given by a map of the land or by a global positioning system. Her altitude is given by the coordinate y. She is walking towards the horizontal position given by x=c. As she does so, she notices that her altitude approaches L. If later asked to guess the altitude over x=c, she would then answer L, even if she had never actually reached that position.

What, then, does it mean to say that her altitude approaches L? It means that her altitude gets nearer and nearer to L. In other words, she gets close to L except for a possible small error in accuracy. For example, suppose we set a particular accuracy goal for our traveler: she must get within a meter of L. She reports back that indeed she can get within one meter: she notes that when she is within five meters from x=c, her altitude is always one meter or less from L. We then change our accuracy goal: can she get within one centimeter? Yes. If she is within seven centimeters of x=c, then her altitude remains within one centimeter of the target L. In summary, to say that the traveler's altitude approaches L as her horizontal position approaches c means that for every target accuracy goal, there is some neighborhood of c whose altitude remains within that accuracy goal.

The initial informal statement can now be explicated

The limit of a function f(x) as x approaches c is a number L with the following property: given any target neighborhood of L, there is a neighborhood of c over which the values of f(x) remain within the target neighborhood.

This explicit statement is quite close to the formal definition of the limit of a function with values in a Hausdorff topological space.

Functions into a Hausdorff space

Let X be a topological space and Y a Hausdorff space. Let c ∈X be a limit point of X and f : X-{c} → Y a function. Then:

L ∈Y is the limit of f(x) as x approaches c if, for every neighborhood V of L, there exists a neighborhood U of c such that f(U-{c}) ⊂ V.

In this case, it is customary to write

\lim_{x\rightarrow c}f(x) = L.

Functions on metric spaces

Suppose f : (M,dM) -> (N,dN) is a map between two metric spaces, p is a limit point of M and L∈N. We say that the limit of f at p is L and write

\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∈M with 0 < dM(x, p) < δ, we have dN(f(x), L) < ε.

Real-valued functions

The real line with metric d(x,y): = | x − y | is a metric space. Also the extended real line with metric d(x,y): = | arctan(x) − arctan(y) | is a metric space.


Limit of a 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

\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

\lim_{x \to \infty}f(x) = \infty

if and only if

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 the expressions

\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 with metric d(x,y): = | x − y | 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

\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 of more than one variable

By noting that |x-p| represents a distance, the definition of a limit can be extended to functions of more than one variable.

\lim_{(x,y) \to (p, q)} f(x, y) = L

if and only if

for every ε > 0 there exists a δ > 0 such that for all real numbers x with 0 < ||(x,y)-(p,q)|| < δ, we have |f(x,y)-L| < ε

where ||(x,y)-(p,q)|| represents the Euclidean distance. This can be extended to any number of variables.

Examples

Real-valued functions

\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 defined at the point, so the limit is equal to the direct evaluation of the function.

\lim_{x \to 0^+}x^x=1

The limit of xx as x approaches 0 from the right is 1.

\lim_{x \to 0}{1 \over x} = \mbox{Undefined}
\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 left is -∞.

\lim_{x \to 0^+}{|x| \over x}=1
\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. Note that |x|/x = -1 if x is negative and |x|/x = 1 if x is positive.

\lim_{x \to 0}x \sin {1 \over x} = 0

The limit of x sin(1/x) as x approaches 0 is 0.

\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.

\lim_{x \to \infty}{x^a \over b^x} = 0 \mbox{ if } a,b \in \mathbb{R}; b>1

Any power function vanishes in magnitude compared to any increasing exponential function as x approaches infinity.

\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.

\lim_{x \to \infty}{a^x \over x!} = 0 \mbox{ if } a \in \mathbb{R}

Any exponential function vanishes in magnitude compared to the 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 (xn) in M with limit equal to p, the sequence (f(xn)) converges with limit L.

If the sets A, B, ... form a finite partition of the function domain, x\in\overline{A} \land x\in\overline{B}, ... and the relative limit for each of those sets exist and is the equal to, say, L, then the limit exists for the point x and is equal to L.

The function f is continuous at p if and only if the limit of f(x) as x approaches p exists and is finite. 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, provided the limits on the right sides of the identity below exist:

\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)/g(x)) & = & {\lim_{x \to p} f(x) / \lim_{x \to p} g(x)} \end{matrix}

(the last provided that the denominator is non-zero). In each case above, when the limits on the right do not exist, or, in the last case, when the limits in both the numerator and the denominator are zero, nonetheless the limit on the left may still exist -- this depends on which functions f and g are.

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×∞, ∞−∞, and ∞/∞ — are also not covered by these rules, but the corresponding limits can often be determined with L'Hôpital's rule.

See also

References