Post on 01-Mar-2016
MATH 38Mathematical Analysis III
I. F. Evidente
IMSP (UPLB)
November 12, 2013
Outline
1 Sets of Numbers
2 Laws on Exponents
3 Review of some limitsRational FunctionsSome transcendental functionsComposition of functionsHandling indeterminate formsSqueeze Theorem
Outline
1 Sets of Numbers
2 Laws on Exponents
3 Review of some limitsRational FunctionsSome transcendental functionsComposition of functionsHandling indeterminate formsSqueeze Theorem
Leonhard Euler (1707-1783)
Some very important numbers
e, i ,pi,1,0
Which are counting numbers?Which are integers?Which are rationals?Which are irrational?Which are real numbers?Which are imaginary?Which are complex?
Some very important numbers
e, i ,pi,1,0
Which are counting numbers?
Which are integers?Which are rationals?Which are irrational?Which are real numbers?Which are imaginary?Which are complex?
Some very important numbers
e, i ,pi,1,0
Which are counting numbers?Which are integers?
Which are rationals?Which are irrational?Which are real numbers?Which are imaginary?Which are complex?
Some very important numbers
e, i ,pi,1,0
Which are counting numbers?Which are integers?Which are rationals?
Which are irrational?Which are real numbers?Which are imaginary?Which are complex?
Some very important numbers
e, i ,pi,1,0
Which are counting numbers?Which are integers?Which are rationals?Which are irrational?
Which are real numbers?Which are imaginary?Which are complex?
Some very important numbers
e, i ,pi,1,0
Which are counting numbers?Which are integers?Which are rationals?Which are irrational?Which are real numbers?
Which are imaginary?Which are complex?
Some very important numbers
e, i ,pi,1,0
Which are counting numbers?Which are integers?Which are rationals?Which are irrational?Which are real numbers?Which are imaginary?
Which are complex?
Some very important numbers
e, i ,pi,1,0
Which are counting numbers?Which are integers?Which are rationals?Which are irrational?Which are real numbers?Which are imaginary?Which are complex?
If time allows, we will prove this
e i x = cosx+ i sinx
If time allows, we will prove this
e i x = cosx+ i sinx
Sets of Numbers
counting numbers:
N= {1,2,3, ...}
integers:Z= {...,2,1,0,1,2, ...}
rationals:Q=
{p
q| p,q Z and q 6= 0
}
irrationals:e,pi,
p2 Q
reals:R=QQ
Sets of Numbers
counting numbers:N= {1,2,3, ...}
integers:Z= {...,2,1,0,1,2, ...}
rationals:Q=
{p
q| p,q Z and q 6= 0
}
irrationals:e,pi,
p2 Q
reals:R=QQ
Sets of Numbers
counting numbers:N= {1,2,3, ...}
integers:
Z= {...,2,1,0,1,2, ...}
rationals:Q=
{p
q| p,q Z and q 6= 0
}
irrationals:e,pi,
p2 Q
reals:R=QQ
Sets of Numbers
counting numbers:N= {1,2,3, ...}
integers:Z= {...,2,1,0,1,2, ...}
rationals:Q=
{p
q| p,q Z and q 6= 0
}
irrationals:e,pi,
p2 Q
reals:R=QQ
Sets of Numbers
counting numbers:N= {1,2,3, ...}
integers:Z= {...,2,1,0,1,2, ...}
rationals:
Q={p
q| p,q Z and q 6= 0
}
irrationals:e,pi,
p2 Q
reals:R=QQ
Sets of Numbers
counting numbers:N= {1,2,3, ...}
integers:Z= {...,2,1,0,1,2, ...}
rationals:Q=
{p
q| p,q Z and q 6= 0
}
irrationals:e,pi,
p2 Q
reals:R=QQ
Sets of Numbers
counting numbers:N= {1,2,3, ...}
integers:Z= {...,2,1,0,1,2, ...}
rationals:Q=
{p
q| p,q Z and q 6= 0
}
irrationals:
e,pi,p2 Q
reals:R=QQ
Sets of Numbers
counting numbers:N= {1,2,3, ...}
integers:Z= {...,2,1,0,1,2, ...}
rationals:Q=
{p
q| p,q Z and q 6= 0
}
irrationals:e,pi,
p2 Q
reals:R=QQ
Sets of Numbers
counting numbers:N= {1,2,3, ...}
integers:Z= {...,2,1,0,1,2, ...}
rationals:Q=
{p
q| p,q Z and q 6= 0
}
irrationals:e,pi,
p2 Q
reals:
R=QQ
Sets of Numbers
counting numbers:N= {1,2,3, ...}
integers:Z= {...,2,1,0,1,2, ...}
rationals:Q=
{p
q| p,q Z and q 6= 0
}
irrationals:e,pi,
p2 Q
reals:R=QQ
Outline
1 Sets of Numbers
2 Laws on Exponents
3 Review of some limitsRational FunctionsSome transcendental functionsComposition of functionsHandling indeterminate formsSqueeze Theorem
Laws on Exponents
1 anam =
an+m
2an
am= anm
3 (ab)n = anbn4
(ab
)n= a
n
bn5 (an)m = anm
Laws on Exponents
1 anam = an+m
2an
am= anm
3 (ab)n = anbn4
(ab
)n= a
n
bn5 (an)m = anm
Laws on Exponents
1 anam = an+m2
an
am=
anm
3 (ab)n = anbn4
(ab
)n= a
n
bn5 (an)m = anm
Laws on Exponents
1 anam = an+m2
an
am= anm
3 (ab)n = anbn4
(ab
)n= a
n
bn5 (an)m = anm
Laws on Exponents
1 anam = an+m2
an
am= anm
3 (ab)n =
anbn
4
(ab
)n= a
n
bn5 (an)m = anm
Laws on Exponents
1 anam = an+m2
an
am= anm
3 (ab)n = anbn
4
(ab
)n= a
n
bn5 (an)m = anm
Laws on Exponents
1 anam = an+m2
an
am= anm
3 (ab)n = anbn4
(ab
)n=
an
bn5 (an)m = anm
Laws on Exponents
1 anam = an+m2
an
am= anm
3 (ab)n = anbn4
(ab
)n= a
n
bn
5 (an)m = anm
Laws on Exponents
1 anam = an+m2
an
am= anm
3 (ab)n = anbn4
(ab
)n= a
n
bn5 (an)m =
anm
Laws on Exponents
1 anam = an+m2
an
am= anm
3 (ab)n = anbn4
(ab
)n= a
n
bn5 (an)m = anm
Let n N(1)n =
{
1 if n is even1 if n is odd
1n ={1 if n is even1 if n is odd
Let n N(1)n =
{1 if n is even
1 if n is odd
1n ={1 if n is even1 if n is odd
Let n N(1)n =
{1 if n is even1 if n is odd
1n ={1 if n is even1 if n is odd
Let n N(1)n =
{1 if n is even1 if n is odd
1n ={
1 if n is even1 if n is odd
Let n N(1)n =
{1 if n is even1 if n is odd
1n ={1 if n is even
1 if n is odd
Let n N(1)n =
{1 if n is even1 if n is odd
1n ={1 if n is even1 if n is odd
Examples
1 (4)(2x)n1
2
(1
5
)n (4x5
)n13
(13
)n (2x3
)n
Outline
1 Sets of Numbers
2 Laws on Exponents
3 Review of some limitsRational FunctionsSome transcendental functionsComposition of functionsHandling indeterminate formsSqueeze Theorem
Limits of Rational Functions
Rational Function
h(x)= f (x)g (x)
where f and g are polynomial functions and g is not the constant zerofunction.
Limits of Rational Functions
Evaluate the following limits:
1 limx
2x3+4x2+26x35x =
1
3
2 limx
2x5+4x2+26x35x =
3 limx
2x3+4x2+26x55x = 0
Limits of Rational Functions
Evaluate the following limits:
1 limx
2x3+4x2+26x35x =
1
3
2 limx
2x5+4x2+26x35x =
3 limx
2x3+4x2+26x55x = 0
Limits of Rational Functions
Evaluate the following limits:
1 limx
2x3+4x2+26x35x =
1
3
2 limx
2x5+4x2+26x35x =
3 limx
2x3+4x2+26x55x = 0
Limits of Rational Functions
Evaluate the following limits:
1 limx
2x3+4x2+26x35x =
1
3
2 limx
2x5+4x2+26x35x =
3 limx
2x3+4x2+26x55x = 0
Limits of Rational Functions
Evaluate the following limits:
1 limx
2x3+4x2+26x35x =
1
3
2 limx
2x5+4x2+26x35x =
3 limx
2x3+4x2+26x55x =
0
Limits of Rational Functions
Evaluate the following limits:
1 limx
2x3+4x2+26x35x =
1
3
2 limx
2x5+4x2+26x35x =
3 limx
2x3+4x2+26x55x = 0
Limits of Rational Functions
Evaluate the following limits:
1 limx
2x3+4x2+26x35x =
1
3
2 limx
2x5+4x2+26x35x =
3 limx
2x3+4x2+26x55x = 0
Limits of Transcendental Functions
1 limxe
x =
2 limx2
x =
0
3 limx lnx =
4 limx0+
lnx =
5 limxArctanx =
pi
2
6 limxArctanx =
pi2
Limits of Transcendental Functions
1 limxe
x =2 lim
x2x =
0
3 limx lnx =
4 limx0+
lnx =
5 limxArctanx =
pi
2
6 limxArctanx =
pi2
Limits of Transcendental Functions
1 limxe
x =2 lim
x2x = 0
3 limx lnx =
4 limx0+
lnx =
5 limxArctanx =
pi
2
6 limxArctanx =
pi2
Limits of Transcendental Functions
1 limxe
x =2 lim
x2x = 0
3 limx lnx =
4 limx0+
lnx =
5 limxArctanx =
pi
2
6 limxArctanx =
pi2
Limits of Transcendental Functions
1 limxe
x =2 lim
x2x = 0
3 limx lnx =
4 limx0+
lnx =
5 limxArctanx =
pi
2
6 limxArctanx =
pi2
Limits of Transcendental Functions
1 limxe
x =2 lim
x2x = 0
3 limx lnx =
4 limx0+
lnx =
5 limxArctanx =
pi
2
6 limxArctanx =
pi2
Limits of Transcendental Functions
1 limxe
x =2 lim
x2x = 0
3 limx lnx =
4 limx0+
lnx =
5 limxArctanx =
pi
2
6 limxArctanx =
pi
2
Recall a theorem...
TheoremIf lim
xa g (x)= L and f is continuous at L, then
limxa f (g (x))
= f (L)
Recall a theorem...
TheoremIf lim
xa g (x)= L and f is continuous at L, then
limxa f (g (x))= f (L)
Examples
1 limxcos
(1
x
)=
1
2 limx ln
( xx+1
)= 0
Examples
1 limxcos
(1
x
)= 1
2 limx ln
( xx+1
)= 0
Examples
1 limxcos
(1
x
)= 1
2 limx ln
( xx+1
)=
0
Examples
1 limxcos
(1
x
)= 1
2 limx ln
( xx+1
)= 0
As a consequence...
CorollaryIf lim
xa g (x)= L and f is continuous everywhere, then
limxa f (g (x))= L
Can you give some examples of functions that are continuous everywhere?
As a consequence...
CorollaryIf lim
xa g (x)= L and f is continuous everywhere, then
limxa f (g (x))= L
Can you give some examples of functions that are continuous everywhere?
Examples
1 limxe
1/x2 =
1
2 limxsin
(2
ex
)= 0
3 limxe
x = 0
Examples
1 limxe
1/x2 = 1
2 limxsin
(2
ex
)= 0
3 limxe
x = 0
Examples
1 limxe
1/x2 = 1
2 limxsin
(2
ex
)=
0
3 limxe
x = 0
Examples
1 limxe
1/x2 = 1
2 limxsin
(2
ex
)= 0
3 limxe
x = 0
Examples
1 limxe
1/x2 = 1
2 limxsin
(2
ex
)= 0
3 limxe
x =
0
Examples
1 limxe
1/x2 = 1
2 limxsin
(2
ex
)= 0
3 limxe
x = 0
Also...
Forms:(e):
limit is (e): limit is 0(ln()): limit is (ln(0+)): limit is (Arctan()): limit is pi2(Arctan()): limit is pi2
Also...
Forms:(e): limit is
(e): limit is 0(ln()): limit is (ln(0+)): limit is (Arctan()): limit is pi2(Arctan()): limit is pi2
Also...
Forms:(e): limit is (e):
limit is 0(ln()): limit is (ln(0+)): limit is (Arctan()): limit is pi2(Arctan()): limit is pi2
Also...
Forms:(e): limit is (e): limit is 0
(ln()): limit is (ln(0+)): limit is (Arctan()): limit is pi2(Arctan()): limit is pi2
Also...
Forms:(e): limit is (e): limit is 0(ln()):
limit is (ln(0+)): limit is (Arctan()): limit is pi2(Arctan()): limit is pi2
Also...
Forms:(e): limit is (e): limit is 0(ln()): limit is
(ln(0+)): limit is (Arctan()): limit is pi2(Arctan()): limit is pi2
Also...
Forms:(e): limit is (e): limit is 0(ln()): limit is (ln(0+)):
limit is (Arctan()): limit is pi2(Arctan()): limit is pi2
Also...
Forms:(e): limit is (e): limit is 0(ln()): limit is (ln(0+)): limit is
(Arctan()): limit is pi2(Arctan()): limit is pi2
Also...
Forms:(e): limit is (e): limit is 0(ln()): limit is (ln(0+)): limit is (Arctan()):
limit is pi2(Arctan()): limit is pi2
Also...
Forms:(e): limit is (e): limit is 0(ln()): limit is (ln(0+)): limit is (Arctan()): limit is pi2
(Arctan()): limit is pi2
Also...
Forms:(e): limit is (e): limit is 0(ln()): limit is (ln(0+)): limit is (Arctan()): limit is pi2(Arctan()):
limit is pi2
Also...
Forms:(e): limit is (e): limit is 0(ln()): limit is (ln(0+)): limit is (Arctan()): limit is pi2(Arctan()): limit is pi2
Examples
1 limx ln
(1
x
)=
2 limx0+
Arctan(lnx)= pi2
Examples
1 limx ln
(1
x
)=
2 limx0+
Arctan(lnx)= pi2
Examples
1 limx ln
(1
x
)=
2 limx0+
Arctan(lnx)=
pi2
Examples
1 limx ln
(1
x
)=
2 limx0+
Arctan(lnx)= pi2
Some indeterminate forms
1 limxx sin
(pix
)=
pi
2 limx
(x+1x
)x= e
Some indeterminate forms
1 limxx sin
(pix
)=pi
2 limx
(x+1x
)x= e
Some indeterminate forms
1 limxx sin
(pix
)=pi
2 limx
(x+1x
)x=
e
Some indeterminate forms
1 limxx sin
(pix
)=pi
2 limx
(x+1x
)x= e
Squeeze Theorem
TheoremSuppose the functions f , g and h are defined on some open interval Icontaining a, except possibly at a itself, and that
f (x) g (x) h(x) for all x in I for which x 6= alimxa f (x)= limxa g (x)= L.
Then limxa g (x)= L.
Squeeze Theorem
TheoremSuppose the functions f , g and h are defined on some open interval Icontaining a, except possibly at a itself, and that
f (x) g (x) h(x) for all x in I for which x 6= alimxa f (x)= limxa g (x)= L.
Then limxa g (x)= L.
Squeeze Theorem
1 limxsinx =
dne
2 limx
sinx
x= 0
3 limx
cosx
x= 0
4 limx
3x+cosx34x =
3
4
Squeeze Theorem
1 limxsinx = dne
2 limx
sinx
x= 0
3 limx
cosx
x= 0
4 limx
3x+cosx34x =
3
4
Squeeze Theorem
1 limxsinx = dne
2 limx
sinx
x=
0
3 limx
cosx
x= 0
4 limx
3x+cosx34x =
3
4
Squeeze Theorem
1 limxsinx = dne
2 limx
sinx
x= 0
3 limx
cosx
x= 0
4 limx
3x+cosx34x =
3
4
Squeeze Theorem
1 limxsinx = dne
2 limx
sinx
x= 0
3 limx
cosx
x=
0
4 limx
3x+cosx34x =
3
4
Squeeze Theorem
1 limxsinx = dne
2 limx
sinx
x= 0
3 limx
cosx
x= 0
4 limx
3x+cosx34x =
3
4
Squeeze Theorem
1 limxsinx = dne
2 limx
sinx
x= 0
3 limx
cosx
x= 0
4 limx
3x+cosx34x =
34
Squeeze Theorem
1 limxsinx = dne
2 limx
sinx
x= 0
3 limx
cosx
x= 0
4 limx
3x+cosx34x =
3
4
Sets of NumbersLaws on ExponentsReview of some limitsRational FunctionsSome transcendental functionsComposition of functionsHandling indeterminate formsSqueeze Theorem