LIMITS 2. LIMITS 2.4 The Precise Definition of a Limit In this section, we will: Define a limit...
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Transcript of LIMITS 2. LIMITS 2.4 The Precise Definition of a Limit In this section, we will: Define a limit...
Let f be a function defined on some open
interval that contains the number a, except
possibly at a itself.
Then, we say that the limit of f(x) as x
approaches a is L, and we write
if, for every number , there is
a number such that
lim ( )x a
f x L
0
0
i 0 thenf ( )x a f x L
PRECISE DEFINITION OF LIMIT Definition 2
Since |x - a| is the distance from x to a and
|f(x) - L| is the distance from f(x) to L, and
since can be arbitrarily small, the definition
can be expressed in words as follows. the distance between f(x)
and L can be made arbitrarily small by taking the distance from x to a sufficiently small (but not 0).
Alternatively, the values of f(x) can be made as close as we please to L by taking x close enough to a (but not equal to a).
lim ( )x a
f x L
PRECISE DEFINITION OF LIMIT – in terms of distance
lim ( )x a
f x L
Therefore, in terms of intervals, Definition 2
can be stated as follows.
for every
(no matter how small is), we can find
such that, if x lies in the open interval
and , then f(x) lies in the open interval .
lim ( )x a
f x L
0 0
,a a
x a ,L L
PRECISE DEFINITION OF LIMIT – in terms of interval
We interpret this statement geometrically
by representing a function by an arrow
diagram as in the figure, where f maps
a subset of onto another subset of .
PRECISE DEFINITION OF LIMIT
Figure 2.4.2, p. 89
The definition of limit states that, if any small
interval is given around L, then
we can find an interval around a
such that f maps all the points in
(except possibly a) into the interval .
( , )L L
,a a
PRECISE DEFINITION OF LIMIT
,a a
( , )L L
Figure 2.4.3, p. 89
If is given, then we draw
the horizontal lines and
and the graph of f.
0 y L y L
PRECISE DEFINITION OF LIMIT – in terms of graph
Figure 2.4.4, p. 89
If , then we can find a number
such that, if we restrict x to lie in the interval
and take , then the curve
y = f(x) lies between
the lines
and .
if such a has been
found, then any smaller
will also work.
lim ( )x a
f x L
0
,a a x a
y L y L
PRECISE DEFINITION OF LIMIT
Figure 2.4.5, p. 89
The three figures show that, if a smaller is chosen, then a smaller may be required.
PRECISE DEFINITION OF LIMIT
Figure 2.4.4, p. 89 Figure 2.4.5, p. 89 Figure 2.4.6, p. 89
Use a graph to find a number
such that
In other words, find a number that corresponds to in the definition of a limit for the function with a = 1 and L = 2.
3if 1 then 5 6 2 0.2x x x
0.2
3( ) 5 6f x x x
PRECISE DEFINITION OF LIMIT Example 1
Rewrite the inequality into graph the curves , y = 1.8, and y = 2.2 near
the point (1, 2).
estimate the x-coordinate of intersections are
about 0.911 and 1.124
Solution: Example 131.8 5 6 2.2x x
Figure 2.4.7, p. 89
3 5 6y x x
So, rounding to be safe, we can say that
This interval (0.92, 1.12) is not symmetric about x = 1.( left distance = 0.08, right distance = 0.12 )
Choose to be the
smaller distance, that is,
Assure the inequality
3if 0.92 1.12 then 1.8 5 6 2.2x x x
Solution: Example 1
Figure 2.4.8, p. 89
0.08
if 1 0.08x
3then 5 6 2 0.2x x
• Let be a given positive number. We want to find a number such that
• However,• Therefore, we want
• That is,
• This suggests that we should choose
if 0 3 then 4 5 7x x
4 5 7 4 12 4 3 4 3x x x x
Proof: Example 2
if 0 3 then 4 3x x
if 0 3 then 34
x x
4
showing that this works. Given , choose . If , then
Thus, Therefore, by the definition of a limit,
0
4
0 3x
4 5 7 4 12 4 3 4 44
x x x
if 0 3 then 4 5 7x x
PROOF Example 2
3
lim 4 5 7x
x
Left-hand limit is defined as follows.
if, for every number , there is
a number such that
Notice that Definition 3 is the same as Definition 2 except that x is restricted to lie in the left half of the interval .
lim ( )x a
f x L
0
0
if then ( )a x a f x L
PRECISE DEFINITION OF LIMIT Definition 3
,a a ,a a
Right-hand limit is defined as follows.
if, for every number , there is
a number such that
In Definition 4, x is restricted to lie in the right half of the interval .
lim ( )x a
f x L
if then ( )a x a f x L
PRECISE DEFINITION OF LIMIT Definition 4
,a a ,a a
0
0
Let be a given positive number.
Here, a = 0 and L = 0, so we want to find a number such that .
That is, . Squaring both sides of the inequality ,
we get . This suggests that we should choose .
if 0 then 0x x
Example 3STEP 1: GUESSING THE VALUE
if 0 thenx x x
2if 0 thenx x 2
Given , let .
If , then .
So, .
According to Definition 4, this shows that
0 2
0 x 2x
0x
0lim 0.x
x
STEP 2: PROOF Example 3
Let be given. We have to find a number such that
To connect with we write
Then, we want
0 0 2if 0 3 then 9x x
STEP 1: GUESSING THE VALUE Example 4
2 9x 3x 2 9 ( 3)( 3)x x x
if 0 3 then 3 || 3x x x
Since
Thus we have
So , if x is chose 1 distance from 3
And
3 1 2 4x x
5 3 7x
3 7x
Example 4STEP 1: GUESSING THE VALUE
3 3 7 3x x x
However, now, there are two restrictions
on , namely
and
To make sure that both inequalities are satisfied, we take to be the smaller of the two numbers 1 and .
The notation for this is .
3x 3 1x 3
7x
C
7
min 1, 7
Example 4STEP 1: GUESSING THE VALUE
Given , let .
If , then (as in part l).
We also have , so
This shows that .
0 min 1, 7
0 3x 3 1 2 4 3 7x x x
3 7x 2 9 3 3 7
7x x x
2
3lim 9xx
STEP 2: PROOF Example 4
Using definition, we prove the
Sum Law. If and both exist, thenlim ( )
x af x L
lim ( )
x ag x M
lim ( ) ( )x a
f x g x L M
PRECISE DEFINITION OF LIMIT
-
Using the Triangle Inequality
we can write:
a b a b
( ) ( ) ( ) ( ( ) ) ( ( ) )
( ) ( )
f x g x L M f x L g x M
f x L g x M
PROOF OF THE SUM LAW Definition 5
We make less than
by making each of the terms
and less than .
Since and , there exists a number such that
Similarly, since , there exists a number such that
( ) ( ) ( )f x g x L M ( )f x L
( )g x M 2
02 lim ( )
x af x L
1 0 1if 0 then ( )
2x a f x L
lim ( )x ag x M
2 0 2if 0 then ( )
2x a g x M
PROOF OF THE SUM LAW
Let .
Notice that
So, and
Therefore, by Definition 5,
1 2if 0 then 0 and 0x a x a x a
( )2
f x L
( )2
g x M
( ) ( ) ( ) ( ) ( )
2 2
f x g x L M f x L g x M
PROOF OF THE SUM LAW
1 2min ,
To summarize,
Thus, by the definition of a limit,
if 0 then ( ) ( ) ( )x a f x g x L M
lim ( ) ( )x a
f x g x L M
PROOF OF THE SUM LAW
Let f be a function defined on some open
interval that contains the number a, except
possibly at a itself.
Then, means that, for every
positive number M, there is a positive
number such that
lim ( )x a
f x
if 0 then ( )x a f x M
INFINITE LIMITS Definition 6
A geometric illustration is shown
in the figure. Given any horizontal line y = M, we can find a number
such that, if we restrict x to lie in the interval but , then the curve y = f(x) lies above the line y = M.
You can see that, if a larger M is chosen, then a smaller may be required.
0 ,a a x a
INFINITE LIMITS
Figure 2.4.10, p. 94
Use Definition 6 to prove that
Let M be a given positive number. We want to find a number such that
However,
So, if we choose and , then .
This shows that as .
20
1lim .x x
INFINITE LIMITS Example 5
2
1if 0 thenx Mx
22
1 1 1M x x
Mx M
1M
10 xM
21 Mx
2
1
x 0x
Let f be a function defined on some open
interval that contains the number a, except
possibly at a itself.
Then, means that, for every
negative number N, there is a positive
number such that
lim ( )x a
f x
if 0 then ( )x a f x N
INFINITE LIMITS Definition 7