| ↔ previous | next ≈ |
Definition (by Heine)
Suppose x0 ∈ R ∪ {-∞, ∞}. Let f be a function defined on a contiguity S(x0) of a point x0. We say that g ∈ R ∪ {-∞, +∞} is the limit of the function f at the point x0, provided that:
for every sequence (xn) of elements of the contiguity S(x0), with limit x0,
g is the limit of a sequence (f (xn)).
The fact that g is the limit of a function f at the point x0 is denoted as
For g ∈ R, the limit is called the finite limit. For g ∈ {-∞, +∞}, we say that the limit is infinite.
It can be shown that if the limit (or one-sided limit) of a function at a point exists then it is unique, i.e., it is not possible that a function has to different limits at a point.
The last definition implies a very useful corollary.
Corollary
If in S(x0) there exist two sequences (an), (bn), both with the limit x0, and the limits of sequences (f (an)), (f (bn)) are different, then the limit of the function f at the point x0 does not exist.
Example
Using the last corollary we will prove that the limit of the function y = sin(1/x) at the point x0 = 0 does not exist. Consider the sequences with general terms
Obviously both have the limit 0, but the squence (sin(1/an)) has the limit 1, whereas the sequence (sin(1/bn)) has the limit 0.
Definition
Suppose x0 ∈ R and f is a function defined (at least) on a left-hand contiguity 
of a point x0.
We say that g ∈ R ∪ {-∞,∞} is a left-hand limit of the function f at the point x0 if for every sequence (xn), of elements of a left-hand contiguity S-(x0), which converges to x0, the sequence (f (xn)) is convergent to the limit g.
The fact that g is the left-hand limit of the function f at the point x0, is denoted as:
Similarly on defines a right-hand limit of a function at a point (denoted as:
The left- and right-hand limits are called one-sided limits.
There is a connection between a limit of a function at a point and
the corresponding one-sided limits:
Theorem
.
The following theorem, called the Catchy definition, can be regarded as a condition equivalent to the Heine definition of the finite limit of a function at a point x0 ∈ R .
Theorem
Let x0, g ∈ R. Suppose f is a function defined on a contiguity S(x0) of a point x0. Then
Similarly, using the "delta-epsilon" notation, one could formulate about 11 conditions equivalent to Heine definitions for one-sided limits, infinite limits, etc. Fortunately this formalism is beyond the scope of this lecture.
The following theorem is frequently used, almost always without referring to it:
Theorem
If for the functions f, g :
then
.
Computing the limit by substitution
one applies exactly the above theorem.
The following theorem is very helpful in calculating the limits, but one has to be careful of getting caught in the trap when the assumptions are not fulfilled, i.e. when, for example, the function is not elementary, or the limit is an indeterminate expression.
Theorem
Suppose f is the elementary function.
Example
Example
Remark
In practice such comments, as in the above examples, are in usually omitted.
Theorem
If
,
where g1, g2 ∈
R, then
if
provided that the left- and right-hand side are definite.
Remark
Similar theorem can be stated for one-sided limits.
The following limits should be remembered:
Theorem
the limits 
and 
can be determined by use of the formula
Example
The following theorem is frequently used by computation of limits
Theorem
If for the functions f, g:
(1) f(x) ≤ g(x) in a contingency S(x0) of a point x0 ∈ R ∪ {-∞, ∞}
Analogous theorem is valid for one-sided limits.
Example
If for the functions f1, f2, f3 the following conditions are fulfilled:
then 
.
Analogous theorem is valid for one-sided limits.
Example
Indeed 2 ≤ E(x) ≤ 4 (∀ x ∈ S(3, 1) = (2, 4))
Thus,
2(x - 3)² ≤ (x - 3)² ≤ 4(x - 3)² (∀ x ∈ S(3, 1) = (2, 4))
Theorem
If a function g (x) is bounded in a neighborhood S (x0) of a point x0 and
Example
Theorem
If there exists a neighborhood S (x0) of a point x0, such that f(x) ≤ g (x) for any x0 ∈ S (x0), then
.
| ↔ previous | next ≈ |