Lecture 3
LIMITS OF FUNCTIONS
CONTINUITY
INTERVALS
INTERVALS
If a and b are real numbers and a ≤ b, then we shall use the
standard interval notation
[a,b] = {x R: a ≤ x ≤ b}
[a,b) = {x R: a ≤ x < b}
(a,b] = {x R: a < x ≤ b}
(a,b) = {x R: a < x < b}.
As you may know, an interval of the form [a,b] is called a
closed interval and an interval of the form (a,b) is called an
open interval. We can also define intervals of infinite length:
If a is any real number, then we define
[a,∞) = {x R: a ≤ x}
(a,∞) = {x R: a < x}
(-∞,a] = {x R: x ≤ a}
(-∞,a) = {x R: x < a},
and, finally, the symbol (-∞,∞) is the set R of all real
numbers.
Note that if a is any real number then the interval (a,a) is the
empty set
Definition
A neighbourhood of point a is an open interval of the form
Definition
A punctured neighbourhood of point a is a set of the form
)
a
,
a
(
)
a
,
a
(
)
a
,
a
(
)
x
(
D
0
for some
0
for some
0
)
,
(
)
(
D
)
,
(
)
(
D
The neighbourhoods of infinity
LIMITS OF FUNCTIONS
Definition (by Heine)
Suppose Let f be a function defined on
a punctured neighbourhood D(x
0
) of a point x
0
.
We say that is the limit of f at x
0
if
for every sequence (x
n
) of elements of the punctured
neighbourhood S(x
0
), tending to x
0
, L is the limit of a
sequence (f (x
n
)) i.e.
L
)
x
(
f
x
x
n
0
n
.
This is denoted by
For the limit is called finite. For
we say that the limit is infinite.
].
,
[
x
0
]
,
[
L
L
)
x
(
f
or
L
)
x
(
f
lim
0
0
x
x
x
x
)
,
(
L
L
,
L
THEOREM
If a function has a limit at a point, then that
limit is unique.
L
)
a
(
f
x
a
n
0
n
M
)
b
(
f
x
b
n
0
n
M
L
If there exist two diferent sequences (a
n
) and (b
n
) both of elements of
the domain of f tending to x
0
, for which
and
with
then the limit of f at x
0
does not exist.
Epsilon-delta definition
Challange: be as close as ε to L,
Response: keep closer than δ to x
0
.
L
)
x
(
f
x
x
,
0
,
0
L
)
x
(
f
lim
0
x
x
0
find such δ
There is no way to be closer then ε (e.g. at point L-ε) to L
(here we have a one side limit).
L
L - ε
One sided limits and limits to infinity
Theorem
PROPERTIES – THE COMBINATION RULES
0
n
provided
x
f
lim
))
x
(
f
(
lim
n
x
x
n
x
x
0
0
)
x
g
(lim
)
x
f
(lim
x
g
x
f
lim
0
0
0
x
x
x
x
x
x
0
x
g
lim
and
0
x
g
if
x
g
lim
x
f
lim
x
g
x
f
lim
0
0
o
0
x
x
x
x
x
x
x
x
x
g
lim
x
f
lim
x
g
x
f
lim
0
0
0
x
x
x
x
x
x
defined
is
)
x
(
f
provided
x
f
lim
)
x
(
f
lim
n
n
x
x
n
x
x
0
0
1
.
2.
3
.
4.
5.
)
,
(
L
],
,
[
x
0
LIMIT RULES
n
n
0
1
n
n
0
1
n
n
x
b
a
b
x
b
x
b
a
x
a
x
a
lim
0
b
x
b
x
b
a
x
a
x
a
lim
0
1
m
m
0
1
n
n
x
m
n
0
1
m
m
0
1
n
n
x
b
a
sgn
b
x
b
x
b
a
x
a
x
a
lim
Theorem
n < m
n > m
n = m
None of these functions has a limit as x tends to 0.
5
2
1
5
x
13
x
1
4
1
x
3
5
lim
x
13
x
x
4
1
x
3
5
lim
x
13
x
x
4
1
x
3
5
lim
13
x
x
4
x
3
x
5
lim
2
x
2
2
x
2
x
2
x
-|x|, x is negative
x
1
sin
)
x
(
f
,
R
)
,
0
(
:
f
)
6
,
001
.
0
(
x
1
2
3
4
5
6
-1
-0.5
0.5
1
)
1
,
001
.
0
(
x
0.2
0.4
0.6
0.8
1
-1
-0.5
0.5
1
A function for which the limit does not exist
zoom in:
)
01
.
0
,
001
.
0
(
x
0.002
0.004
0.006
0.008
0.01
-1
-0.5
0.5
1
x
1
sin
)
x
(
f
,
R
)
,
0
(
:
f
Computing the limit by substitution
)
y
(
f
lim
y
y
x
x
)
x
(
g
y
))
x
(
g
(
f
lim
0
0
y
y
0
0
x
x
THE SQUEEZE PRINCIPLE
Let I be an interval containing the point a. Let f, g, and
h be functions defined on I, except possibly at a itself.
Suppose that for every
we have:
and also suppose that:
Then
.
a
x
x
,
I
g (x) < f (x) < h (x)
L
)
(
lim
)
(
lim
x
h
x
g
a
x
a
x
L
)
(
lim
x
f
a
x
The graph of f is squeezed between g and h.
?
)
x
2
x
sin
x
(
lim
x
x
x
2
x
x
2
x
sin
x
x
lim
x
)
x
2
x
sin
x
(
lim
x
Find
?
)
x
2
x
sin
x
(
lim
x
x
2
x
sin
x
x
2
x
x
)
x
x
sin
x
2
(
lim
x
Find
Any function whose graph is squeezed
between
1+ x
2
/2 and 1- x
2
/4 has a limit 1 as x-
> 0.
Theorem
If a function g(x) is bounded in a neighborhood O(a)
of a point a and
bounded
0
)
(
)
(
lim
then
,
0
)
(
lim
x
g
x
f
x
f
a
x
a
x
A consequence of the Squeeze Theorem:
0.2
0.4
0.6
0.8
1
1.2
1.4
-0.2
0.2
0.4
0.6
0.8
x
1
sin
x
)
x
(
f
,
R
)
,
0
(
:
f
0.005
0.01
0.015
0.02
-0.015
-0.01
-0.005
0.005
0.01
0.015
0.02
)
2
,
0
(
x
)
02
.
0
,
0
(
x
1
x
x
sin
lim
x
x
sin
)
x
(
f
,
R
)
,
(
:
f
0
x
USEFULL LIMIT
Sketch of the
proof of
1
sin
lim
0
tan
so
,
1
but
,
tan
TA
OA
TA/OA
limit
hand
right
2
0
.
1
OAT
OAP
OAP
Δ
Area
tor
sec
circular
Area
Area
2
r
2
1
tor
sec
circular
of
Area
cos
1
sin
1
sin
2
1
tan
2
1
2
1
sin
2
1
Sinθ is even, so the left-hand limit exists and
has the same value as the right-hand limit.
1
1
1
sin
lim
0
1
x
x
sin
lim
..
4
0
x
e
x
1
1
lim
.
3
x
x
e
y
y
y
1
0
1
lim
a
n
x
e
x
a
1
lim
0
x
tan
c
arc
lim
,
x
tan
c
arc
lim
,
2
x
arctan
lim
,
2
x
arctan
lim
.
5
x
x
x
x
x
ln
lim
,
x
ln
lim
.
6
x
0
x
1
a
0
for
1
a
for
0
a
lim
1
a
0
for
0
1
a
for
a
lim
.
2
x
x
x
x
0
a
for
0
x
1
lim
x
lim
0
a
for
,
x
lim
.
1
a
x
a
x
a
x
IMPOTRANT
LIMITS
ASYMPTOTES
The line y = a is a HORIZONTAL ASYMPTOTE for f if
a
x
f
)
(
lim
x
a
x
f
)
(
lim
x
The line x = a is a VERTICAL ASYMPTOTE of a
function f if either of the following conditions is true:
)
x
(
lim
a
x
f
)
x
(
lim
a
x
f
SLANT = OBLIQUE = TILTED ASYMPTOTE:
If y = m x + b, is any non-vertical line, then the function f (x) is asymptotic to it if
]
)
(
[
lim
)
(
lim
x
x
mx
x
f
b
x
x
f
m
0
)
(
lim
0
)
(
lim
x
x
b
mx
x
f
b
mx
x
f
-4
-2
2
4
-4
-2
2
4
-40
-20
20
40
-40
-20
20
40
mx+b
Slant
asypto
te
Two Slant Asymptotes
A curve can intersect its asymptote, even infinitely many times.
CONTINUITY
)
x
(
O
0
)
x
(
O
0
Continuous from the right at a point x
0
, if there exists a right-hand neighborhood
of the point x
0
which is contained in the domain D
f
and
.
Continuous from the left at a point x
0
, if there exists a left-hand neighborhood
of the point x
0
which is contained in the domain D
f
and
Definition
A function f is said to be:
Definition
A function f is said to be:
Continuous at a point x
0
, if there exists a neighborhood
O(x
0
) of the point x
0
which is contained in the domain D
f
and
.
Theorem
A function f is continuous at point x
0
if BOTH one-sided limits at x
0
exist
and
)
x
(
f
)
x
(
f
lim
0
x
x
0
)
x
(
f
)
x
(
f
lim
)
x
(
f
lim
0
x
x
x
x
0
0
Theorem
If the functions f and g are continuous at a point x
0
, then the functions:
1. f + g, f - g
2. f g
3. a f and a f + bg, where
4. f / g (provided g (x
0
) ≠ 0)
are continuous at x
0
as well.
R
b
,
a
Definition
Let be an interval.
If P = (a, b) (is open), then we say that a function f is
continuous on P if f is continuous at each point of P.
If P = [a, b] then we say that a function f is continuous
on P if f is continuous from the right at a, continuous
on the left at b and continuous at each point of an open
interval Int P = (a, b).
If P = [a, b) then we say that a function f is continuous
on P if f is continuous from the right at a, and
continuous at each point of an open interval Int P = (a,
b). Likewise is defined the continuity of a function f on
(a,b].
f
D
P
Theorem (on continuity of a composition)
If functions f and g are continuous at points x
0
and f (x
0
), respectively,
then a compound function g (f (x)) is continuous at x
0
.
Theorem (on the continuity of inverse functions)
If the domain D
f
of a function f is an interval and f is
either strictly decreasing or increasing and continuous,
then
(a) the range f (D
f
) of f is an interval
(b) the inverse function f
-1
exists and is continuous in its
domain f(D
f
).
CONTINUITY OF ELEMENTARY FUNCTIONS!!!
Definition
The basic elementary functions are: constant, power (incl.
root), exponential, logarithmic, trigonometric and inverse
trigonometric functions.
The elementary functions are basic elementary functions
and all functions built up of a finite combination of basic
elementary functions, arithmetic operations using the four
elementary operations (+ – × ÷). and compositions of
functions.
•Every polynomial w(x) = a
n
x
n
+ a
n-1
x
n-1
+...+a
1
x+a
0
is the elementary function
•Every rational function, i.e. the ratio of two polynomials, is an elementary function
•The equality
implies that g(x) = x
tgx
is an elementary function.
•The equality implies that h(x) = | x | is an elementary function.
)
x
ln(tan
x
)
x
(
g
,
e
)
x
(
f
where
),
x
)(
g
f
(
x
x
x
tan
x
x
2 2
Theorem
Elementary functions are continuous in every interval
contained in their domain.
THEOREMS
Theorem (the intermediate value theorem)
If y = f(x) is continuous on [a,b], f (a) ≠ f (b), and u is a
number between f(a) and f(b), then there is at least one c ,
between a < c < b such that
f(c) = u.
x
y
a
c
b
f(a)
f(b)
u
f(x)
Corollary
If a function f is continuous in a closed interval [a, b] and
f (a) f (b) < 0, then there exists a point such
that f (c) = 0.
This corollary is frequently used to demonstrate the
existence of a root of an equation in a given interval.
]
b
,
a
[
c
The theorem represents the idea that the graph of a
continuous function on a closed interval can be drawn without
lifting your pencil from the paper.
c
2
b
y = f(x)
a
=
c
1
max
min
Theorem (Extreme Value Theorem - Weierstrass)
If a function f is continuous on a closed interval [a, b],
then there exists two points c
1
, c
2
in [a,b] such that
f(c
1
) is the global minimum of f on [a,b] i.e. f (c
1
) ≤ f (x)
and such that f( c
2
) is the global maximum of f on
[a,b] i.e. f (x) ≤ f (c
2
)
This theorem means, that the range of f ([a, b]) of a function f
continuous in an interval [a, b] contains the smallest element
f(c
1
) and the largest element f(c
2
).
MONOTONE
FUNCTIONS
strictly increasing
strictly decreasing
i
n
c
r
e
a
s
i
n
g
increasing
decreasing
MONOTONE FUNCTIONS
We say that the function f is increasing if whenever t and x
belong to S and for t < x we have f(t) ≤ f(x).
The function f is strictly increasing if whenever t and x
belong to S and
t < x we have f(t) < f(x).
A function that is either increasing or decreasing is said to
be monotone, and a function that is either strictly increasing
or strictly decreasing is said to be strictly monotone.
We say that the function f is decreasing if whenever t and x
belong to S and for t < x we have f(t) < f(x).
The function f is strictly decreasing if whenever t and x
belong to S and
t < x we have f(t) > f(x).
Suppose that S is a set of real numbers and that f : S → R.
We see at once that a strictly monotone function is
always one-one. However, a one-one function does not
have to be strictly monotone.
EXTREMA
MINIMUM, MAXIMUM
A function has a global maximum at x
0
, if f(x
0
) ≥ f(x) for all x from the domain
A function has a global minimum at x
0
, if f(x
0
) < f(x) for all x from the domain
An extremum is a maximum or minimum.
An extremum may be local ( i.e. a relative extremum; an
extremum in a given region which is not the overall maximum
or minimum) or global (absolute).
Extrema: minimum,
maximum
A real-valued function f defined on the real line is said to have
a local maximum at the point x
0
,
if there exists some ε > 0, such that f(x
0
) ≥ f(x) when |x − x
0
| <
ε.
The value of the function at this point is called local maximum
of the function.
A real-valued function f defined on the real line is said to have
a local minimum at the point x
0
,
if there exists some ε > 0, such that f(x
0
) < f(x) when |x − x
0
| <
ε.
The value of the function at this point is called local minimum
of the function.
Functions with many extrema can be very dificult to graph.
Pathological Examples
This function f(x) = sin(e
2x+9)
has 16 480 extrema
in the closed interval [0,1]
PECULIAR FUNCTIONS
THE POPCORN FUNCTION
This function was originally defined by the mathematician Johannes Thomae
http://demonstrations.wolfram.com/TheModifiedDirichletFunction/
Theta, little-oh, big-Oh
Θ(.), o(.), O(.),
A theoretical measure of the execution of an algorithm, usually
the time or memory needed, given the problem size n, which is
usually the number of items.
RATE OF GROWTH
A big confusion in notations
f(n) = Θ (g(n))
means there are positive constants c
1
, c
2
,
and k, such that
0 ≤ c
1
g(n) ≤ f(n) ≤ c
2
g(n)
for all n ≥ k.
The values of c
1
, c
2
, and k must be fixed for the function f and must
not depend on n.
read: „ f is Theta of g ".
)
,
0
(
,
const
)
n
(
g
)
n
(
f
lim
)
n
(
g
(
)
n
(
f
n
Often Θ is confused with O
The same rate of growth
f(n) = O(g(n))
read: „ f is
big Oh of g ".
means there are positive constants c and k, such that
0 ≤ f(n) ≤ cg(n) for all n
≥ k.
The values of c and k must be fixed for the function f and must not
depend on n.
2n
2
= O(n
3
)
(c = 1, k =
2)
not really an equality, should be
)
n
(
O
n
2
3
2
Often big-Oh is treated like Theta
f(n) is at most the order of g(n)
f(n) = o(g(n))
means for all c > 0 there exists some k > 0 such that
0 ≤ f(n) < c g(n) for
all n ≥ k.
The value of k must not depend on n, but may depend on c.
g(n) grows much faster
than f(n).
,
0
)
n
(
g
)
n
(
f
lim
)
n
(
g
(
o
)
n
(
f
n
When the limits exist:
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