Special Limits
definition of e
The number e is defined as a limit. Here is one definition:
1
e = lim (1 + x) x
x→0+
Special Limits
definition of e
The number e is defined as a limit. Here is one definition:
1
e = lim (1 + x) x
x→0+
A good way to evaluate this limit is make a table of numbers.
Special Limits
definition of e
The number e is defined as a limit. Here is one definition:
1
e = lim (1 + x) x
x→0+
A good way to evaluate this limit is make a table of numbers.
x
1
(1 + x) x
.1
.01
0.001
0.0001
0.00001
→0
Special Limits
definition of e
The number e is defined as a limit. Here is one definition:
1
e = lim (1 + x) x
x→0+
A good way to evaluate this limit is make a table of numbers.
x
1
(1 + x) x
.1
2.5937
.01
0.001
0.0001
0.00001
→0
Special Limits
definition of e
The number e is defined as a limit. Here is one definition:
1
e = lim (1 + x) x
x→0+
A good way to evaluate this limit is make a table of numbers.
x
1
(1 + x) x
.1
2.5937
.01
2.70481
0.001
0.0001
0.00001
→0
Special Limits
definition of e
The number e is defined as a limit. Here is one definition:
1
e = lim (1 + x) x
x→0+
A good way to evaluate this limit is make a table of numbers.
x
1
(1 + x) x
.1
2.5937
.01
2.70481
0.001
2.71692
0.0001
0.00001
→0
Special Limits
definition of e
The number e is defined as a limit. Here is one definition:
1
e = lim (1 + x) x
x→0+
A good way to evaluate this limit is make a table of numbers.
x
1
(1 + x) x
.1
2.5937
.01
2.70481
0.001
2.71692
0.0001
2.71814
0.00001
→0
Special Limits
definition of e
The number e is defined as a limit. Here is one definition:
1
e = lim (1 + x) x
x→0+
A good way to evaluate this limit is make a table of numbers.
x
1
(1 + x) x
.1
2.5937
.01
2.70481
0.001
2.71692
0.0001
2.71814
0.00001
2.71826
→0
Special Limits
definition of e
The number e is defined as a limit. Here is one definition:
1
e = lim (1 + x) x
x→0+
A good way to evaluate this limit is make a table of numbers.
x
1
(1 + x) x
.1
2.5937
.01
2.70481
Where e = 2.7 1828 1828 · · ·
0.001
2.71692
0.0001
2.71814
0.00001
2.71826
→0
→e
Special Limits
definition of e
The number e is defined as a limit. Here is one definition:
1
e = lim (1 + x) x
x→0+
A good way to evaluate this limit is make a table of numbers.
x
1
(1 + x) x
.1
2.5937
.01
2.70481
0.001
2.71692
0.0001
2.71814
Where e = 2.7 1828 1828 · · ·
This limit will give the same result:
e = lim
x→∞
1+
1
x
x
0.00001
2.71826
→0
→e
Special Limits
alternate definition of e
Here is an equivalient definition for e:
1 x
e = lim 1 +
x→∞
x
Special Limits
alternate definition of e
Here is an equivalient definition for e:
1 x
e = lim 1 +
x→∞
x
x
1
(1 + x) x
100
1000
10000
1000000
→∞
Special Limits
alternate definition of e
Here is an equivalient definition for e:
1 x
e = lim 1 +
x→∞
x
x
1
(1 + x) x
100
2.70481
1000
10000
1000000
→∞
Special Limits
alternate definition of e
Here is an equivalient definition for e:
1 x
e = lim 1 +
x→∞
x
x
1
(1 + x) x
100
2.70481
1000
2.71692
10000
1000000
→∞
Special Limits
alternate definition of e
Here is an equivalient definition for e:
1 x
e = lim 1 +
x→∞
x
x
1
(1 + x) x
100
2.70481
1000
2.71692
10000
2.71815
1000000
→∞
Special Limits
alternate definition of e
Here is an equivalient definition for e:
1 x
e = lim 1 +
x→∞
x
x
1
(1 + x) x
100
2.70481
1000
2.71692
10000
2.71815
1000000
2.71827
→∞
Special Limits
alternate definition of e
Here is an equivalient definition for e:
1 x
e = lim 1 +
x→∞
x
x
1
(1 + x) x
100
2.70481
1000
2.71692
10000
2.71815
1000000
2.71827
→∞
→e
Special Limits
alternate definition of e
Here is an equivalient definition for e:
1 x
e = lim 1 +
x→∞
x
x
1
(1 + x) x
100
2.70481
1000
2.71692
10000
2.71815
Note that
lim f(x)
x→0
is the same as
1000000
2.71827
→∞
→e
Special Limits
alternate definition of e
Here is an equivalient definition for e:
1 x
e = lim 1 +
x→∞
x
x
1
(1 + x) x
100
2.70481
1000
2.71692
10000
2.71815
Note that
lim f(x)
x→0
is the same as
lim f
x→∞
1
x
1000000
2.71827
→∞
→e
Special Limits
e the natural base
I
the number e is the natural base in calculus. Many
expressions in calculus are simpler in base e than in other
bases like base 2 or base 10
Special Limits
e the natural base
I
the number e is the natural base in calculus. Many
expressions in calculus are simpler in base e than in other
bases like base 2 or base 10
I
e = 2.71828182845904509080 · · ·
Special Limits
e the natural base
I
the number e is the natural base in calculus. Many
expressions in calculus are simpler in base e than in other
bases like base 2 or base 10
I
e = 2.71828182845904509080 · · ·
I
e is a number between 2 and 3. A little closer to 3.
Special Limits
e the natural base
I
the number e is the natural base in calculus. Many
expressions in calculus are simpler in base e than in other
bases like base 2 or base 10
I
e = 2.71828182845904509080 · · ·
I
e is a number between 2 and 3. A little closer to 3.
I
e is easy to remember to 9 decimal places because 1828
repeats twice: e = 2.718281828. For this reason, do not use
2.7 to extimate e.
Special Limits
e the natural base
I
the number e is the natural base in calculus. Many
expressions in calculus are simpler in base e than in other
bases like base 2 or base 10
I
e = 2.71828182845904509080 · · ·
I
e is a number between 2 and 3. A little closer to 3.
I
e is easy to remember to 9 decimal places because 1828
repeats twice: e = 2.718281828. For this reason, do not use
2.7 to extimate e.
I
use the ex button on your calculator to find e. Use 1 for x.
Special Limits
e the natural base
I
the number e is the natural base in calculus. Many
expressions in calculus are simpler in base e than in other
bases like base 2 or base 10
I
e = 2.71828182845904509080 · · ·
I
e is a number between 2 and 3. A little closer to 3.
I
e is easy to remember to 9 decimal places because 1828
repeats twice: e = 2.718281828. For this reason, do not use
2.7 to extimate e.
I
use the ex button on your calculator to find e. Use 1 for x.
I
example: F = Pert is often used for calculating compound
interest in business applications.
Special Limits
f(x) =
1
x
Here are four useful limits:
I
lim
x→+∞
1
x
→ 0+ = 0
Special Limits
f(x) =
1
x
Here are four useful limits:
I
lim
x→+∞
I
lim
x→−∞
1
x
1
x
→ 0+ = 0
→ 0− = 0
Special Limits
f(x) =
1
x
Here are four useful limits:
I
lim
x→+∞
I
lim
x→−∞
I
lim
x→0+
1
x
1
x
1
x
→ 0+ = 0
→ 0− = 0
→ +∞
Special Limits
f(x) =
1
x
Here are four useful limits:
I
lim
1
x
x→+∞
I
lim
1
x→−∞
I
lim
x→0+
I
lim
x→0−
x
1
x
1
x
→ 0+ = 0
→ 0− = 0
→ +∞
→ −∞
Special Limits
f(x) =
1
x
cont.
The general idea is this:
I
1
+BIG
→ +small
Special Limits
f(x) =
1
x
cont.
The general idea is this:
I
1
+BIG
I
1
−BIG
→ +small
→ −small
Special Limits
f(x) =
1
x
cont.
The general idea is this:
I
1
+BIG
I
1
−BIG
I
→ +small
→ −small
1
+small
→ +BIG
Special Limits
f(x) =
1
x
cont.
The general idea is this:
I
1
+BIG
I
1
−BIG
I
→ +small
→ −small
1
+small
I
1
−small
→ +BIG
→ −BIG
Special Limits
f(x) =
1
x
f
+9.0
+8.0
+7.0
+6.0
+5.0
+4.0
+3.0
+2.0
+1.0
−8.5
−7.0
−5.5
−4.0
−2.5
−1.0
−1.0
−2.0
−3.0
−4.0
−5.0
−6.0
−7.0
−8.0
−9.0
+1.0
+2.5
+4.0
+5.5
+7.0
+8.5
x
Special Limits
1
BIG
= small,
1
small
= BIG
Look at 1x using some numbers:
x
10 100 1000 10000
f(x) = 1x
.1 .01 .001 .0001
x
f(x) =
1
x
-10
-.1
x
f(x) =
1
x
.1
10
x
f(x) =
1
x
-.1
-10
-100
-.01
.01
100
-.01
-100
-1000
-.001
.001
1000
-.001
-1000
100000
.00001
-10000
-.0001
.0001
10000
-100000
-.00001
.00001
100000
-.0001
-10000
→ +∞
→ 0+
→ −∞
→ −0+
→ +0
→ +∞
-.00001
-100000
→ −0
→ −∞
Special Limits
x → ±∞ for Polynomials
I
when taking limits of polynomials to ±∞ drop the lower
degree terms and only keep the higest degree term of of the
polymomial.
Special Limits
x → ±∞ for Polynomials
I
when taking limits of polynomials to ±∞ drop the lower
degree terms and only keep the higest degree term of of the
polymomial.
I
this is an intermediate step in taking the limit. Use algebra to
simplify the expression at this step then continue to work on
finding the limit to infinity.
Special Limits
x → ±∞ for Polynomials
I
when taking limits of polynomials to ±∞ drop the lower
degree terms and only keep the higest degree term of of the
polymomial.
I
this is an intermediate step in taking the limit. Use algebra to
simplify the expression at this step then continue to work on
finding the limit to infinity.
I
this works because for large values of x the highest power
term of the polynomial is so much larger that all of the
smaller degree terms that the smaller degree terms have no
effect in the limit to infinity.
Special Limits
examples of x → ±∞ for Polynomials
I
limx→+∞
2x3 +99x+1000
1+2x2 +4x3
Special Limits
examples of x → ±∞ for Polynomials
I
limx→+∞
2x3 +99x+1000
1+2x2 +4x3
Special Limits
examples of x → ±∞ for Polynomials
I
limx→+∞
2x3 +99x+1000
1+2x2 +4x3
= limx→+∞
2x3
4x3
Special Limits
examples of x → ±∞ for Polynomials
I
limx→+∞
2x3 +99x+1000
1+2x2 +4x3
= limx→+∞
2x3
4x3
= limx→+∞
2
4
Special Limits
examples of x → ±∞ for Polynomials
I
2x3 +99x+1000
1+2x2 +4x3
1
limx→+∞ 2
limx→+∞
= limx→+∞
2x3
4x3
= limx→+∞
2
4
=
Special Limits
examples of x → ±∞ for Polynomials
I
2x3 +99x+1000
1+2x2 +4x3
1
limx→+∞ 2 = 12
limx→+∞
= limx→+∞
2x3
4x3
= limx→+∞
2
4
=
Special Limits
example of x → ±∞ for Polynomials
limx→+∞
2x3 +99x+1000
1+2x2 +4x3
Special Limits
example of x → ±∞ for Polynomials
limx→+∞
2x3 +99x+1000
1+2x2 +4x3
= limx→+∞
2x3
4x3
Special Limits
example of x → ±∞ for Polynomials
limx→+∞
2x3 +99x+1000
1+2x2 +4x3
= limx→+∞
2x3
4x3
= limx→+∞
2
4
Special Limits
example of x → ±∞ for Polynomials
limx→+∞
2x3 +99x+1000
1+2x2 +4x3
= limx→+∞
2x3
4x3
= limx→+∞
2
4
= limx→+∞
1
2
Special Limits
example of x → ±∞ for Polynomials
limx→+∞
2x3 +99x+1000
1+2x2 +4x3
= limx→+∞
2x3
4x3
= limx→+∞
2
4
= limx→+∞
1
2
=
1
2
Special Limits
example of x → ±∞ for Polynomials
limx→−∞
2x4 +99x+1000
1+2x2 +4x3
Special Limits
example of x → ±∞ for Polynomials
limx→−∞
2x4 +99x+1000
1+2x2 +4x3
= limx→−∞
2x4
4x3
Special Limits
example of x → ±∞ for Polynomials
limx→−∞
2x4 +99x+1000
1+2x2 +4x3
= limx→−∞
2x4
4x3
= limx→−∞ 42 x
Special Limits
example of x → ±∞ for Polynomials
limx→−∞
2x4 +99x+1000
1+2x2 +4x3
= limx→−∞
2x4
4x3
= limx→−∞ 42 x
= limx→−∞ 21 x
Special Limits
example of x → ±∞ for Polynomials
limx→−∞
2x4 +99x+1000
1+2x2 +4x3
= limx→−∞
2x4
4x3
= limx→−∞ 42 x
= limx→−∞ 21 x
=
1
2
limx→−∞ x
Special Limits
example of x → ±∞ for Polynomials
limx→−∞
2x4 +99x+1000
1+2x2 +4x3
= limx→−∞
2x4
4x3
= limx→−∞ 42 x
= limx→−∞ 21 x
=
1
2
limx→−∞ x
=
1
2
· −∞
Special Limits
example of x → ±∞ for Polynomials
limx→−∞
2x4 +99x+1000
1+2x2 +4x3
= limx→−∞
2x4
4x3
= limx→−∞ 42 x
= limx→−∞ 21 x
=
1
2
limx→−∞ x
=
1
2
· −∞
= −∞
Special Limits
example of x → ±∞ for Polynomials
limx→+∞
2x4 +99x+1000
1+2x2 +4x6
Special Limits
example of x → ±∞ for Polynomials
limx→+∞
2x4 +99x+1000
1+2x2 +4x6
= limx→+∞
2x4
4x6
Special Limits
example of x → ±∞ for Polynomials
limx→+∞
2x4 +99x+1000
1+2x2 +4x6
= limx→+∞
2x4
4x6
= limx→+∞
2
4x2
Special Limits
example of x → ±∞ for Polynomials
limx→+∞
2x4 +99x+1000
1+2x2 +4x6
= limx→+∞
2x4
4x6
= limx→+∞
2
4x2
= limx→+∞
1
2x2
Special Limits
example of x → ±∞ for Polynomials
limx→+∞
2x4 +99x+1000
1+2x2 +4x6
= limx→+∞
2x4
4x6
= limx→+∞
2
4x2
= limx→+∞
1
2x2
=
1
2
limx→+∞
1
x2
Special Limits
example of x → ±∞ for Polynomials
limx→+∞
2x4 +99x+1000
1+2x2 +4x6
= limx→+∞
2x4
4x6
= limx→+∞
2
4x2
= limx→+∞
1
2x2
=
1
2
limx→+∞
=
1
2
·
1
+∞
1
x2
Special Limits
example of x → ±∞ for Polynomials
limx→+∞
2x4 +99x+1000
1+2x2 +4x6
= limx→+∞
2x4
4x6
= limx→+∞
2
4x2
= limx→+∞
1
2x2
=
1
2
limx→+∞
1
x2
=
1
2
·
=
1
2
· 0+ = 0+ = 0
1
+∞
Special Limits
example of x → ±∞ for Polynomials
limx→+∞
2x4 +99x+1000
1+2x2 +4x6
= limx→+∞
2x4
4x6
= limx→+∞
2
4x2
= limx→+∞
1
2x2
=
1
2
limx→+∞
1
x2
=
1
2
·
=
1
2
· 0+ = 0+ = 0
1
+∞
Special Limits
example of x → ±∞ for Polynomials
limx→+∞
2x4 +99x+1000
1+2x2 +4x6
= limx→+∞
2x4
4x6
= limx→+∞
2
4x2
= limx→+∞
1
2x2
=
1
2
limx→+∞
1
x2
=
1
2
·
=
1
2
· 0+ = 0+ = 0
1
+∞
Special Limits
example of x → ±∞ for Polynomials
limx→+∞
2x4 +99x+1000
1+2x2 +4x6
= limx→+∞
2x4
4x6
= limx→+∞
2
4x2
= limx→+∞
1
2x2
=
1
2
limx→+∞
1
x2
=
1
2
·
=
1
2
· 0+ = 0+ = 0
1
+∞
Special Limits
example of x → ±∞ for Polynomials
limx→+∞
2x4 +99x+1000
1+2x2 +4x6
= limx→+∞
2x4
4x6
= limx→+∞
2
4x2
= limx→+∞
1
2x2
=
1
2
limx→+∞
1
x2
=
1
2
·
=
1
2
· 0+ = 0+ = 0
1
+∞
Special Limits
example of x → ±∞ for Polynomials
limx→+∞
2x4 +99x+1000
1+2x2 +4x6
= limx→+∞
2x4
4x6
= limx→+∞
2
4x2
= limx→+∞
1
2x2
=
1
2
limx→+∞
1
x2
=
1
2
·
=
1
2
· 0+ = 0+ = 0
1
+∞
Special Limits
example of x → ±∞ for Polynomials
limx→+∞
2x4 +99x+1000
1+2x2 +4x6
= limx→+∞
2x4
4x6
= limx→+∞
2
4x2
= limx→+∞
1
2x2
=
1
2
limx→+∞
1
x2
=
1
2
·
=
1
2
· 0+ = 0+ = 0
1
+∞