Limit
Limit of a scalar function from a single variable
Denote the open set D ⊂ R and the function f : D &rr; R of one variable. The point x0 is the limit point of the set D: x0 ∈ D. The limit of the function f for x tending to x0 is h if for any ε > 0 there is such a δ > 0 that
x ∈ D, x≠x0, |x-x0| < δ ∴ |f(x)-h| < ε
(∴ stands for "hence")
One-way limit
The right-hand limit of f when x tends to x0 on the right is equal to h1 if for any ε > 0 exists δ > 0 such that
x ∈ D, x > x0, |x-x0| < δ ∴ |f(x)-h1| < ε
The left-hand limit of f when x tends to x0 on the left is h2 if for any ε > 0 exists δ > 0 such that
x ∈ D, x < x0, |x-x0| < δ ∴ |f(x)-h2| < ε
If there is a limit f at the point x0 and its value is equal to h, then there are one-sided limits and they match the value of h. If the one-way limits do not match, then the limit does not exist
Properties of limits
If there are limits of functions f and g at point a, then
- limx→a(c⋅f(x)) = c⋅limx→a(f(x))
- limx→a(f(x)+g(x)) = limx→a(f(x)) + limx→a(g(x))
- limx→a(f(x)⋅g(x)) = limx→a(f(x)) ⋅ limx→a(g(x))
- limx→a(f(x)/g(x)) = limx→a(f(x)) / limx→a(g(x)) (if limx→a(g(x))≠0)
- limx→a(f(x)g(x)) = limx→a(f(x)) limx→a(g(x))
Limit of a function with two variables
Let D⊂R2 be an open set and a function f: D→R is a function with two variables. Point a is the limit point. The limit of the function f when x tends to a is equal to h if for any ε > 0 there exists such a value δ > 0 that
x ∈ D, x ≠ a, ||x-a|| < δ ∴ |f(x)-h| < ε
Limit definition
Let D be an open subset of Rn and the function f : D→ Rm. Point a∈ D and h∈ Rm. The limit of a function f when x tends to a is equal to h if for any ε > 0 there exists δ > 0 such that
x ∈ D, x ≠ a, ||x-a|| < δ ∴ |f(x)-h| < ε
and is written as:
limx→af(x) = h
Limit Properties
Point a∈D does not necessarily belong to the set D, therefore, may not exist, i.e. for the existence of a limit at point a there is no need for the function to be defined at this point.
f(x) approaches h when x approaches a, in other words, the function tends to h when x tends to a: f(x) → h.
If there is a limit limx→af(x), then it is the only one, that is, it cannot exist two different limits at one point.
Filter limit
Let's give a function f: D ⊂ Rn → Rm and a subset of S ⊂ D such that a ∈ S. Filter limit S at point a is equal to h if for any ε > 0 there exists such a value σ > 0 that:
x ∈ S, x ≠ a, ||x-a|| < δ ∴ ||f(x)-h|| < ε
If the limit of the function at any point is equal to h, then the value of the limit for any filter S will also be equal to h. Thus, if there is no limit on at least one filter, then, therefore, the limit does not exist. Also, if the limits on the two filters do not match, then there is no limit.
Example
Let the function f be given: R2\{(0,0)} → R
f(x,y)= xy/(x2+y2)
Denote two filters S1 = {(x,y)∈R2 / x = y} and S2 = {(x,y)∈R2 / x = -y} and find the limits at point 0:
x∈S1: lim[(x,y)→(0,0)]f(x,y)=lim [x→0]x2/2x2=1/2
x∈S2: lim[(x,y)→(0,0)]f(x,y)=lim [x→0]-x2/2x2=-1/2
The limits don't match, so the limit doesn't exist.
Continuity of the function
The function f is given: D ⊂ Rn → Rm with the domain of definition D and the point a ∈ D. A function f is continuous at point a if lim x→af(x)=f(a). A function is called continuous if it is continuous at any point from the domain of definition D.
Properties of continuous functions
For functions f: D⊂ Rn → Rm and g: D ⊂ Rn → Rm and the numbers c∈ R are fair:
- If f is continuous at point a, then the function c⋅f is also continuous
- If f and g are continuous at point a, then the function g+f is also continuous
- If f and g are continuous at point a and m=1, then the function f⋅g is also continuous at point a
- If f⊂ Rn→ R is continuous at point a and is not zero in the entire domain of definition, then the function 1/f is also continuous at point a
- if f⊂ Rn→ Rm and f(x) = (f1(x),...,fm(x)), then a function f is continuous at point x if and only if all functions fi(x) are continuous for 1≤ i≤ m
Continuity of complex functions
Given the functions f:D⊂Rn→Rm and g:V⊂Rm→Rp. Subject to the conditions f(D)⊂V and g(f(x)), if f is continuous at point a, and g is continuous at point b = f(a), then g(f(x)) is also continuous at point a.
The Cantor-Heine theorem
Let be given a function f : D⊂ Rn → R m and a domain D such that D is a compact set. If the function is continuous, then the function there must be a minimum and maximum value.
