limit algebraic rules
DESCRIPTION
Limit Algebraic Rules. Rule 1 – Plug the x xalue into the equation If you get a/b and b 0, the answer is a/b b = 0 and a 0, the answer is d.n.e you get 0/0 then go to higher rules. 0.0 0.1. Limit Algebraic Rules. Rule 2 – Factor and cancel return to Rule 1 - PowerPoint PPT PresentationTRANSCRIPT
Limit Algebraic RulesLimit Algebraic Rules
Rule 1 – Plug the x xalue Rule 1 – Plug the x xalue into the equationinto the equation
If you get a/b andIf you get a/b and b 0, the answer is a/bb 0, the answer is a/b b = 0 and a 0, the answer is d.n.eb = 0 and a 0, the answer is d.n.e you get 0/0 then go to higher rulesyou get 0/0 then go to higher rules
..
0.00.0
0.10.1
2
3
9lim
1x
x
x
Limit Algebraic RulesLimit Algebraic Rules
Rule 2 – Factor and cancelRule 2 – Factor and cancel return to Rule 1return to Rule 1
SOAP (aSOAP (a33 + b + b33) = (a + b)(a) = (a + b)(a22 – ab + b – ab + b22)) Diff of 2 squares (aDiff of 2 squares (a22 - b - b22) = (a + b)(a – b)) = (a + b)(a – b) a xa x22 + bx + c Find two numbers whose + bx + c Find two numbers whose
product is ac, but add to bproduct is ac, but add to b
..
12.012.0
0.10.1
2
2( 2)( )
2i
2 4l mx
x x x
x
Rule 3 – Conjugate Rule 3 – Conjugate
3
2 15l
3im
3
x
x
x 3
2 15 3
3
2 15 3lim
2 15 3
x
x
x
x
x
_20.3333
3 3
3
2 15 9 1lim
2 15 33
x x
x
x 3
3 2
3lim
2 15 3x
x
x x
. 4. 4
4.04.0
0.10.1 2
2
2 2
2 2lim
2 2x
x
x
x
x
2
2 2 2lim
12 4x
x
x
x
Limits as x goes to Limits as x goes to + or - infinity+ or - infinity
What does y get close to What does y get close to
as x goes to + infinity?as x goes to + infinity? What does y get close to What does y get close to
as x goes to - infinity?as x goes to - infinity?
Limits as x goes to Limits as x goes to + or - infinity+ or - infinity
To calculate the limit as xTo calculate the limit as x
goes to infinity, remembergoes to infinity, remember
if n > 0.if n > 0.
2
2 2 2
2
2 2 2
4 3 7
l1
m2
i
x
x xx x xx xx x x
2
2
4 3 7
2l
1im
x
x x
x x
2
2
3 7
lim4
2 11
x
x x
x x
lim1
0nx x
44
1
Limits as x goes to Limits as x goes to + or - infinity+ or - infinity
2 2
2
2 2 2
lim
4 3
2 2x
xx x
x xx x x
2
4 3i
2 2l mx
x
x x
2
2
4
im2
1l
3
2x
x x
x x
lim1
0nx x
00
1
Limits as x goes to Limits as x goes to + or - infinity+ or - infinity
3
3 3
2
3 3 3
4 3
li2
m2x
x xx x
x xx x x
3
2
4 3i
2 2l mx
x x
x x
2
2 3
34
1 2 2limx
x
x x x
lim1
0nx x
. . .d n e
55
f(x)=3xf(x)=3x5 5 - 3x- 3x3 3 + 2+ 2 g(x) = 2xg(x) = 2x2 2 - 4x + 2- 4x + 2
( )l m
( )ix
f x
g x
( )l m
( )ix
g x
f x
55
f(x)=3xf(x)=3x5 5 - 3x- 3x3 3 + 2+ 2 g(x) = 2xg(x) = 2x5 5 - 4x + 2- 4x + 2
( )l m
( )ix
f x
g x
( )l m
( )ix
g x
f x
55
f(x)=3xf(x)=3x5 5 - 3x- 3x3 3 + 2+ 2 g(x) = 2xg(x) = 2x21 21 - 4x + 2- 4x + 2
( )l m
( )ix
f x
g x
( )l m
( )ix
g x
f x
55
f(x)= xf(x)= x8 8 - 3x- 3x3 3 + 2+ 2 g(x) = 7xg(x) = 7x8 8 - 4x + 2- 4x + 2
( )l m
( )ix
f x
g x
( )l m
( )ix
g x
f x
Evaluate .Evaluate .
A.A. 33
B.B. 1/3 1/3
C.C. d.n.ed.n.e
D.D. 00
2
2
3 4 2lim
3x
x x
x
Evaluate .Evaluate .
A.A. 33
B.B. 1/3 1/3
C.C. d.n.ed.n.e
D.D. 00
2
2
4 2lim
3 3x
x x
x
Evaluate .Evaluate .
A.A. 33
B.B. 1/3 1/3
C.C. d.n.ed.n.e
D.D. 00
4
2
4 2lim
3 3x
x x
x
Evaluate .Evaluate .
A.A. 33
B.B. 1/3 1/3
C.C. d.n.ed.n.e
D.D. 00
4
6
4 2lim3 3 1x
x x
x x
A.A. 00
B.B. d.n.e.d.n.e.
C.C. 77
2
2
9 3 7
2limx
x x
x
A.A. 9/89/8
B.B. d.n.e.d.n.e.
C.C. 00
1 1
9 7 4 9 7 9 7 4lim lim
9 7 4
4
1 1x x
x x x
x x x
x, if 0 < xx, if 0 < x-x, if x < 0-x, if x < 0
2 24 3 7lim
/
(2 2) /x
x x x
x x
41
2
2 | | {x x
24 3 7
2l m
2ix
x x
x
2
2 2 2
4
2l
3
2im
7
x
x xx x xxx x
lim1
0nx x
x, if 0 < xx, if 0 < x-x, if x < 0-x, if x < 0
2
2 2 2
4 3 7
2 2limx
x xx x xxx x
24 3 7
2l
2imx
x x
x
2 24 3 7 /
(l
2 2im
) /( )x
x x x
x x
41
2
2 | | {x x
..
3.03.0
0.10.1
2 2 29 3 7 9 3 7 /
2 (li
) /m
2lim
x x
x x x x x
x x x
..
-3.0-3.0
0.10.1
2 2 29 3 7 9 3 7 /
2 ( 2) /lim li
)m
(x x
x x x x x
x x x