LIMITS
18
LIMITS
description
LIMITS. If , find the following value of x in given table. What is Limits?. A limit is the maximum or minimum value that a given function. A limit will equal or approach (get closer to). Example: - PowerPoint PPT Presentation
Transcript of LIMITS
LIMITS
If , find the following value of x in given table. 2
4)(
2
x
xxf
x → 1 from left x→ 1 from right
x 0,8 0,9 0,99 0,999 → 1 → 1,0001 1,001 1,01 1,05 1,1
F(x) … … …. …. → … → ……… …….. …… …… …..
from left, f(x) approach to … from right, f(x) approach to …
What is Limits?
A limit is the maximum or minimum value that a given function.
A limit will equal or approach (get closer to).Example:
a car can get faster and faster; however, at some point it will reach the maximum speed that it can possibly go. Therefore we say that the car has reached it’s maximum speed limit.
Back to previous table. The answer is:
What can you deduce about that function?
x → 1 from left x→ 1 from right
x 0,8 0,9 0,99 0,999 → 1 → 1,0001 1,001 1,01 1,05 1,1
F(x) 2,8 2,9 2,99 2,999 → 3 → 3,0001 3,001 3,01 3,05 3,1
from left, f(x) approach to 3 from right, f(x) approach to 3
That means:Limit f(x) if x approach to 1, and if writes to determine the value of limit, we can choose x ε R that value of x apprroach whether form left or right of x.
So:
2
4)(
2
x
xxf
32
4x
1
2
xx
Limit
3 Methods to Calculate Limit of Function
1. Table of Limit2. Substitution
if x = 1 substitute to , then f(x) = 3. This
result is the same with value of that limit function. example: find of this following limit:
a) b)
c)
2
4)(
2
x
xxf
However, in several problems, we can not do substitution method.
Example: find the value of
If x = 2 substitute to , the answer is .
It means that the limit doesn’t have value.The problems must calculate with different
method.
2
4x
2
2
xx
Limit
2
4)(
2
x
xxf
0
0
3. Factorization
example: find this following limits:
a.
b.
4)2(22
2)-2)(x(x
22
4x
2
2
x
x
Limit
xx
Limit
xx
Limit
1x
1-x 2
1xlim
2-x
65x-x 2
2xlim
1
4x
2 3
2
xx
Limit 2
82x
2 2
2
xx
x
x
Limit
4
82x
4
2
x
x
x
Limit
Determine the value of limit using rationalize
Simplify the square root of function.example: simplify !answer:
31 x
10
91
913131
)31).(31(31
x
x
xxx
xxx
one more example: simplify !
answer:
10
31
x
x
)31(
1
)31(10
10
)31(10
9)1(31
31.
10
31
10
31
x
xx
x
xx
xx
x
x
x
x
x
Determine limit function with rationalizing.Find the value of : !
Though we’ve already know the simple form of
, then we can substitute x = 10 to
10
31
10
x
x
x
Limit
10
31
x
x
)31(
1
x
The answer:
6
1
3110
131
1
1010
31
10
xx
Limit
x
x
x
Limit
Exercise: find the value of this following limit a.
b.
c.
3x
9-xlim
9x
2-x
14x-3lim
2x
3x
9-xlim
9x
Limit Function For x Approach to Infinity
A. Form .
example: find the value of limit !answer :
=
g(x)
f(x)lim
x
32
23
x 2x6x-3
74xxlim
32
23
x 2x6x-3
74xxlim
2
1
2
1
200
001
2x
6
x
3x
1
x
41
x
2x
x
6x
x
3x
7
x
4x
x
x
3
3
x
3
3
3
2
3
33
2
3
3
xlimlim
exercise:
form the example, we can deduce that:
..............2x10xx-3
210x34x5xlimlim
x32
234
x
x
Limit Function For x Approach to Infinity
B. Form .
To find the value of this limit, we have to multiply
the limit with , and the form will be:
=
=
)()(limx
xgxf
)(f(x)
)(f(x)
xg
xg
)(f(x)
)(f(x).)(f(x)lim
x xg
xgxg
g(x)f(x)
(x)g(x)f 22
xlim
from previous sentence, we can deduce that:
example: find the value of this following limit.a.
b.
510x94x 22
xlim
xx
9xx 2
xlim
