objectives: to evaluate limits numerically, graphically, and analytically. to evaluate infinite...
TRANSCRIPT
Objectives:
• To evaluate limits numerically, graphically, and analytically.
• To evaluate infinite limits
*3-1: Limits
x .9 .99 .999 1 1.0001
1.001 1.01
f(x)
*USING YOUR CALCULATORS, MAKE A TABLE OF VALUES TO FIND THE VALUE THAT f(x) IS APPROACHING AS x IS APPROACHING 1 FROM THE LEFT AND FROM THE RIGHT.
f(x) = 3x + 1
1x 1x“As x is approaching 1 from the left”
“As x is approaching 1 from the right”
1. What do we know about the graph?
2. What does the graph look like near x =1?
1
1)(
2
x
xxf
X .9 .99 .999 1 1.0001
1.001 1.01
* Informal Definition of a Limit
If the values of f(x) approach the number L as x approaches a from both the left and the right, we say that the limit L as x approaches a exists and
**You can use a table of values to find a limit by taking values of x very, very, very close to a on BOTH sides and see if they approach the same value
Lxfax
)(lim
*Something weird….
*A limit describes how the outputs of a function behave as the inputs approach some particular value. It is NOT necessarily the value of the function at that x value (but it could be).
WHAT???????????????? Yes, this is true
*One-Sided LimitsRIGHT-HAND LIMIT (RHL)
(The limit as x approaches a from the right)
LEFT-HAND LIMIT(LHL)
(The limit as x approaches a from the left)
)(lim xfax
)(lim xfax
IN ORDER FOR A LIMIT TO EXIST, THE FUNCTION HAS TO BE APPROACHING THE SAME VALUE FROM BOTH THE LEFT AND THE RIGHT (LHL = RHL)
= )(lim xf
ax )(lim xf
ax
* Let’s take a look at limits graphically!!
)(lim.8
)2(.7
)(lim.6
)(lim.5
)(lim.4
)1(.3
)(lim.2
)(lim.1
2
2
2
1
1
1
xf
f
xf
xf
xf
f
xf
xf
x
x
x
x
x
x
*Example continues…
)(lim.8
)4(.7
)(lim.6
)(lim.5
)(lim.4
)3(.3
)(lim.2
)(lim.1
4
4
4
3
3
3
xf
f
xf
xf
xf
f
xf
xf
x
x
x
x
x
x
)(lim2
xfx
*Graph the following function. Then find the limit.
,12
,8
,
)(
2
x
x
xf
3
3
3
x
x
x
)(lim3
xfx
Look at a table of values and the graph of
What happens as x approaches 2?
DOES NOT EXIST
)(lim2
xfx
)(lim2
xfx
)(lim2
xfx
2
4)(
2
x
xxf
is not a number. It is used to describe a situation where something increases or decreases without bound (gets more and more negative or more and more positive)
A LIMIT DOES NOT EXIST (DNE) WHEN:
1. The RHL and LHL as x approaches some value a are BOTH or BOTH - . We write
or , but the limit DNE.
2. The RHL as x approaches some value a is and the LHL as x approaches the same value is - or vice versa.
3. LHL ≠ RHL
(The fancy dancy explanations are on page 154)
)(lim xfax
)(lim xfax
)(lim.9
)(lim.8
)(lim.7
)(lim.6
)(lim.5
)(lim.4
)(lim.3
)(lim.2
)(lim.1
:
3
3
3
5
5
5
0
0
3
xf
xf
xf
xf
xf
xf
xf
xf
xf
Evaluate
x
x
x
x
x
x
x
x
x
*PRIZE ROUND
Find all the zeros:
2x3+x2-x
*Properties of LimitsIf L, M, a and k are real numbers and and , then
1. Sum/Difference Rule:
2. Product Rule:
3. Constant Multiple Rule:
4. Quotient Rule:
5. Power Rule:
Lxfax
)(lim Mxgax
)(lim
0,))((lim
0,)(
)(lim
))((lim
))()((lim
))()((lim
sLxf
MM
L
xg
xf
Lkxfk
MLxgxf
MLxgxf
s
r
s
r
ax
ax
ax
ax
ax
*Note:
*If one of the limits for one of the functions DNE when using the properties, then the limit for the combined function DNE.
* Other important properties and limits…..
1. If p(x) is a polynomial, then
2. , where c is a constant
3.
)()(lim apxpax
ccax
lim
1sin
lim0
x
xx
*Take a look at p. 165 # 25 and 30.
* How to Find Limits Algebraically1. Try substitution (If a is in the domain of the function this works). If you get 0/0 when you substitute, there is something you can do to simplify!!
2. If substitution doesn’t work, simplify, if possible. Then evaluate limit.
3. Conjugate Multiplication: If function contains a square root and no other method works, multiply numerator and denominator by the conjugate. Simplify and evaluate.
You can always use a table or a graph to reinforce your conclusion
*Prize---Prize---PrizeFactor the following:
1. x3-27
2. 8x3+1
3. 4x2-9
* Finding Limits Algebraically Worksheet--Classwork
Lets do some examples together, shall we????
Handout—Finding Limits Algebraically—Classwork
I do #1,3,5,8,10,11 with you
You try #2,4,7,9
*Some trickier examplesEvaluate the limit:
xx
x
x
x
x
x
x
x
cossinlim.3
sin6lim.2
3113
lim.1
0
0
3
*Evaluate the limit:
2,3
2,1lim.2
4
2lim.1
2
2
4
xx
xxx
x
x
x
x
*Examples: Evaluate the Limit
)3(lim.5
4lim.4
)(lim.3
)(lim.2
lim.1
2
3
3
2
xx
x
xx
xx
x
x
x
x
x
x
Given:
*What is the function’s value approaching as the x values get larger and larger in the positive direction?
Larger and larger in the negative direction?
)(lim
)(lim
xf
xf
x
x
xxf
1)(
FOR ANY POSITIVE REAL NUMBER n AND ANY REAL NUMBER c :
and
TO FIND THE FOR ANY RATIONAL FUNCTION , DIVIDE NUMERATOR AND DENOMINATOR BY THE VARIABLE EXPRESSION WITH THE LARGEST POWER IN DENOMINATOR.
0lim nx x
c0lim
nx x
c
xlim
*Rational Function Examples:
13
52lim.3
13
52lim.2
13
52lim.1
2
3
2
2
2
x
x
x
x
x
x
x
x
x
*Prize
WHEN WE ARE EVALUATING THESE LIMITS AS
x ±∞, WHAT ESSENTIALLY ARE WE FINDING?
*WE LEARNED IT IN PRE-CALC WHEN WE GRAPHED RATIONAL FUNCTIONS
*WHAT DOES THE END-BEHAVIOR OF A FUNCTION TELL US?
*IT BEGINS WITH AN “H”
*This is Calculus!!!! Woohoo!!
*DEFINTION OF HORIZONTAL ASYMPTOTE
*THE LINE y=b IS A HORIZONTAL ASYMPTOTE OF THE GRAPH OF y=f(x) IF EITHER
OR
bxfx
)(limbxf
x
)(lim
* Examples: a.)Evaluate the Limitb.) What is the equation for the HA?
1
2lim.3
1
2lim.2
2
3lim.1
2
3
2
2
2
x
x
x
x
x
x
x
x
x
*Extra Examples, if needed.
4
23
2
5
2
2
2
32
3
2
4lim.5
35lim.4
145
23lim.3
123
45lim.2
5
143lim.1
x
xx
x
x
xx
xx
xx
xx
x
xx
x
x
x
x
x
0;0.5
;.4
5
3;5
3.3
;.2
0;0.1
:
y
none
y
none
y
Answers
***NOTE: A function can have more than one horizontal asymptote. Take a look at these graphs.
3
12
12
232
x
xy
x
xy
* Guidelines for finding limits as x ±∞ of Rational Functions
1. If the degree of the numerator is less than the degree of the denominator, the limit of the rational function is 0.
2. If the degree of the numerator is = to the degree of the denominator, the limit of the rational function is the ratio of the leading coefficients.
3. If the degree of the numerator is greater than the degree of the denominator, then the limit of the rational function approaches ±∞.
0sin
lim x
xx
xx
x
x
xx
x
x
EXAMPLES
x
x
x
22
sinlim.3
sin6lim.2
sin5lim.1
:
*VERTICAL ASYMPTOTES AND INFINITE LIMITS
*LOOK AT THE GRAPH OF
2
3)(
x
xf
)(lim
)(lim
2
2
xf
xf
x
x
*Vertical Asymptote: DefinitionThe line x=a is a vertical asymptote of y=f(x) if either:
OR
)(lim xfax
)(lim xfax
* Properties of Infinite LimitsLet c and L be real numbers and let f and g be functions such that
1. Sum/difference:
2. Product:
3. Quotient:
Lxgxfcxcx
)(lim,)(lim
0)(
)(lim
0,)]()([lim
0,)]()([lim
)]()([lim
xf
xg
Lxgxf
Lxgxf
xgxf
cx
cx
cx
cx
* Find the vertical asymptote. Prove using a limit.
3. Evaluate the limit:
4
82)(.2
12
1)(.1
2
2
x
xxxg
xxf
1
3lim
2
1
x
xxx