from sec 2.1 : find: our goal sec 2.2: limit of function and limit laws sec 2.3: calculating limits...
TRANSCRIPT
21
1lim
2
1
x
xx
From sec 2.1 :
?)(lim
xfax
Find:OUR GOAL
Sec 2.2: Limit of Function and Limit Laws
Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS
Example: Find ??2lim 2
2
xx
x
We say that the limit of f(x) as x approaches 2 from the right is equal to 4
The symbol x 2+ means that we consider only values of x > 2
4)(lim2
xfx
We say that the limit of f(x) as x approaches 2 from the left is equal to 4
The symbol x 2- means that we consider only values of x < 2
4)(lim2
xfx
Sec 2.2: Limit of Function and Limit Laws
Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS
42lim 2
2
xx
x 42lim 2
2
xx
x 42lim 2
2
xx
x
Sec 2.2: Limit of Function and Limit Laws
Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS
?)(lim0
tHx
1)(lim0
tHx
0)(lim0
tHx
exsitnot does )(lim0
tHx
Sec 2.2: Limit of Function and Limit Laws
Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS
Sec 2.2: Limit of Function and Limit Laws
Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS
For the function whose graph is given below
Determine the following limits
)(lim)1
xfax
)(lim)1
xfbx
)(lim)1
xfcx
1
2
DNE
)(lim)2
xfdx
)(lim)2
xfex
)(lim)2
xffx
1
1
1
)(lim)0
xfgx
)(lim)0
xfhx
)(lim)0
xfix
0
0
0
)(lim)5.0
xfjx
)(lim)5.0
xfkx
)(lim)5.0xfl
x
5.0
5.0
5.0
Sec 2.2: Limit of Function and Limit Laws
Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS
Exam-ITerm-122
Sec 2.2: Limit of Function and Limit Laws
Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS
?)(lim
xfax
Find:
1 Create a table: watch the behavior
2 Use the Graph to find the limit
Sec 2.2: Limit of Function and Limit Laws
Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS
Example: Find ??2lim 2
2
xx
x
4)(lim2
xfx
Sec 2.2: Limit of Function and Limit Laws
Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS
)(lim)14x
x
Example:
)100
(lim)2200 x
xx
)200(lim)1400x
Example:
)(lim)22
x
)10(lim)11
x
x
Example:
)5(lim)2 2
2xx
x
12
2lim)3
3
2
0
x
xxx
3
3
2
0 12
2lim)4
x
xxx
234lim)5 2
1
xx
x
Sec 2.2: Limit of Function and Limit Laws
Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS
-
Can we use Direct Substitution Prop.??
Sec 2.2: Limit of Function and Limit Laws
Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS
Can we use Direct Substitution Prop.??
1sin
lim0
x
xx
1sin
lim0
x
xx
1sin
lim0
x
xx
Sec 2.2: Limit of Function and Limit Laws
Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS
1
1)(
2
x
xxf
1
1)(
x
xg
12
11
1)( 2
xif
xifx
xxf
Sec 2.2: Limit of Function and Limit Laws
Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS
?)(lim
xfax
Find:
1 Create a table: watch the behavior
2 Use the Graph to find the limit
3 Given the function f(x)
Sec 2.2: Limit of Function and Limit Laws
Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS
Exam-1 Term 1221)Common Factors
2)Conjugate radical
0
0case
Exam-1 Term 121
Example
1
1lim)1
2
1
x
xx
Example:
Sec 2.2: Limit of Function and Limit Laws
Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS
1)Common Factors
2)Conjugate radical
0
0case
Sec 2.2: Limit of Function and Limit Laws
Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS
2
1
)1(
1lim
)1)(1(
1lim
1
1lim
1121
xxx
x
x
xxxx
1
1)(
2
x
xxf
1
1)(
x
xg
5.0)(lim1
xfx
5.0)(lim1
xgx
5.0)1( gudefined )1(f
What is the difference ??
Sec 2.2: Limit of Function and Limit Laws
Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS
Sec 2.2: Limit of Function and Limit Laws
Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS
Sec 2.2: Limit of Function and Limit Laws
Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS
??|][|lim3x
x
3|][|lim3x
x
2|][|lim3x
x
existnot does |][|lim3xx
Sec 2.2: Limit of Function and Limit Laws
Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS
xx
sinlim0
xx
coslim0
0
1
Sec 2.2: Limit of Function and Limit Laws
Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS
Squeeze theorem = Sandwich theorem
Sec 2.2: Limit of Function and Limit Laws
Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS
)1
sin(lim 2
0x xx
Example:
Sec 2.2: Limit of Function and Limit Laws
Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS
Exam1-091
Sec 2.2: Limit of Function and Limit Laws
Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS
Limit from one side does not exist
exsitnot does )(lim0
tHx
Find an example for a function such that the limit from right does not exist
Sec 2.2: Limit of Function and Limit Laws
Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS
)(lim )1
xgax
)(lim )1
xgbx
)(lim )1xgc
x
)(lim )2
xgdx
)(lim )2
xgex
)(lim )2xgf
x
)2( ) gg
)(lim )'5
xgdx
)(lim )'5
xgex
)(lim )'5xgf
x
)5( )' gg
Sec 2.2: Limit of Function and Limit Laws
Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS
c
1
4)(xg
)(xf
23
Sec 2.2: Limit of Function and Limit Laws
Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS
Sec 2.2: Limit of Function and Limit Laws
Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS
Sec 2.2: Limit of Function and Limit Laws
Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS
Sec 2.2: Limit of Function and Limit Laws
Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS
Sec 2.2: Limit of Function and Limit Laws
Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS
Sec 2.2: Limit of Function and Limit Laws
Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS