section 5.2.3 day 2 limits review
DESCRIPTION
Section 5.2.3 Day 2 Limits Review. Lesson Objective: Students will: Understand the necessary conditions for a limit to exist. Practice finding limits. Predict limits from tables. ( 1). General Idea: Behavior of a function very near the point where - PowerPoint PPT PresentationTRANSCRIPT
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Section 5.2.3 Day 2Limits Review
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Lesson Objective:
Students will:
• Understand the necessary conditions for a limit to exist.
• Practice finding limits.• Predict limits from tables.
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(1). General Idea: Behavior of a function very near the point where
(2). Layman’s Description of Limit (Local Behavior)
L If x approaches a, from both sides, then the
function approaches a single number, L.
a
(3). Notation
(4). Mantra
x ax a
lim ( )x a
f x L
, then .If x a y L
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G N A W
4 Ways to Depict Limits
Graphically
Numerically
Analytically
Words
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G N A W Graphically
2x
Lim f x
1x
Lim f x
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FINDING LIMITS
2( ) 2 1f x x x 3 22 2 4( )
2x x xf x
x
• Graphically 2x
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G N A W
4
28xLim
x
3.9 3.99 3.999 4 4.001 4.01 4.1
7
284
Mantra:
• Numerically
• Words
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FINDING LIMITS
• Analytically
2
3
93x
xLimx
23
39x
xLimx
Rem: Always start with Direct Substitution
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FINDING LIMITS
• Analytically
0( )
xLimf x
1 if 0 ( ) 3x-1 if x 0
x xIff x
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(1). If a is in the domain: Use Substitution
x a
1lim( 3)x
22
lim(8 4)x
x
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(2). If a is not in the domain: Start with Substitution
x a
2
3
9lim 3x
xx
3, then 6.If x y
If the result is , ( this step must be shown),
then factor and substitute again.
00
3
( 3)( 3)lim 3x
x xx
A.
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Graphically
Since the function is undefined at x = 3, this produces a “hole” in the graph.
x
y
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(2). If a is not in the domain: Start with Substitution
x a
23
3lim 9x
xx
3, then .If x y
If the result is , ( this step must be shown),
then factor and substitute again.
#0
3
3lim ( 3)( 3)x
xx x
B.
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Graphically
Since the function is undefined at x = 3, this produces a “hole” and an asymptote in the graph.
x
y
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, when x = odd #, then:ny x
Polynomials (Power Functions)
, when x = even #, then:ny x
lim ( )lim ( )
x
x
f xf x
®¥
®- ¥
=+¥=- ¥
lim ( )lim ( )
x
x
f xf x
®¥
®- ¥
=+¥=+¥
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Remember:
Rational Functions
1lim 0x x®¥
=
#lim 0x x®¥
=
#lim 0nx x®¥=
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Polynomial:Polynomial
Rational Functions
Divide all terms in the numerator & denominator by the largest degree in the denominator.
2
24 3 5) 7 5 1x xa yx x
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Rational FunctionsDivide all terms in the numerator & denominator by the largest degree in the
denominator.2
33 1) 5 2xb yx x
34 5 1) 7 2x xc y
x
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Leading Term Test
Take the ratio of the leading terms.
) If a number, then that number is the limit. The end behavior is an horizontal asymptote #.a
y
) If x is left in the denominator, then the limit is 0. The end behavior is and horizontal asymptote 0. b
y
c) If x is in the numerator, then there is no limit. The end behavior mimics the polynomial.
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Examples
1) lim ( 1)( 2)x
xx x
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Examples
2
28 26 152) lim 2 15x
x xx x
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Examples
2 93) lim 2x
xx
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AssignmentWS 13.1 #7-32