7.4 Relative Rates of Growth
Domination
This section is about comparing functions to see which
dominate as x → ∞.
Definition
Let f (x) and g(x) be positive for some sufficiently large x.
1
f grows faster than g as x → ∞ if
lim
x→ ∞
f (x)
=∞
g(x)
Domination
This section is about comparing functions to see which
dominate as x → ∞.
Definition
Let f (x) and g(x) be positive for some sufficiently large x.
1
f grows faster than g as x → ∞ if
lim
x→ ∞
2
f (x)
=∞
g(x)
f grows slower than g as x → ∞ if
lim
x→ ∞
f (x)
=0
g(x)
Domination
This section is about comparing functions to see which
dominate as x → ∞.
Definition
Let f (x) and g(x) be positive for some sufficiently large x.
1
f grows faster than g as x → ∞ if
lim
x→ ∞
2
f grows slower than g as x → ∞ if
lim
x→ ∞
3
f (x)
=∞
g(x)
f (x)
=0
g(x)
f grows at the same rate as g as x → ∞ if
lim
x→ ∞
f (x)
=L
g(x)
Which Grows Faster?
Example
Which grows faster: ex or x?
Which Grows Faster?
Example
Which grows faster: ex or x?
Hint: Use L’Hopital’s Rule
Which Grows Faster?
Example
Which grows faster: ex or x?
Hint: Use L’Hopital’s Rule
ex grows faster than x
Which Grows Faster?
Example
Which grows faster: ex or x20 ?
Which Grows Faster?
Example
Which grows faster: ex or x20 ?
ex grows faster than x20
Which Grows Faster?
Example
Which grows faster: ex or x20 ?
ex grows faster than x20
Can we generalize this?
Which Grows Faster?
Example
Which grows faster: 2x or 4x ?
Which Grows Faster?
Example
Which grows faster: 2x or 4x ?
2x grows slower than 4x
Which Grows Faster?
Example
Which grows faster: 2x or 4x ?
2x grows slower than 4x
Can we generalize this?
Which Grows Faster?
Example
Which grows faster: ln(x) or x?
Which Grows Faster?
Example
Which grows faster: ln(x) or x?
ln(x) grows slower than x
Which Grows Faster?
Example
Which grows faster: log3 x or log2 x?
Which Grows Faster?
Example
Which grows faster: log3 x or log2 x?
log3 (x) grows at the same rate as log2 (x)
Which Grows Faster?
Example
Which grows faster: log3 x or log2 x?
log3 (x) grows at the same rate as log2 (x)
log3 (x) =
log(x)
log(3)
Which Grows Faster?
Example
Which grows faster: log3 x or log2 x?
log3 (x) grows at the same rate as log2 (x)
log(x)
log(3)
log(x)
log2 (x) =
log(2)
log3 (x) =
Which Grows Faster?
Example
Which grows faster: log3 x or log2 x?
log3 (x) grows at the same rate as log2 (x)
log(x)
log(3)
log(x)
log2 (x) =
log(2)
log3 (x) =
log(x)
log(3)
lim
x→∞ log(x)
log(2)
=
Which Grows Faster?
Example
Which grows faster: log3 x or log2 x?
log3 (x) grows at the same rate as log2 (x)
log(x)
log(3)
log(x)
log2 (x) =
log(2)
log3 (x) =
log(x)
log(3)
lim
x→∞ log(x)
log(2)
=
log(2)
log(3)
Comparisons
Sometimes it is easier to show functions grow at the same rate
by comparing them to a common function.
Comparisons
Sometimes it is easier to show functions grow at the same
√ rate
by comparing
them
to
a
common
function.
Consider
x2 + 1
√
3
3
vs. 2x + 1. Differentiating would be a pain, so we wouldn’t
want to use L’Hopital’s Rule, but we could use something
simple, line x.
√
lim
x→ ∞
x2 + 1
=
x
Comparisons
Sometimes it is easier to show functions grow at the same
√ rate
by comparing
them
to
a
common
function.
Consider
x2 + 1
√
3
3
vs. 2x + 1. Differentiating would be a pain, so we wouldn’t
want to use L’Hopital’s Rule, but we could use something
simple, line x.
r
√
1
x2 + 1
x2
+ 2 =1
= lim
lim
2
x→ ∞
x→ ∞
x
x
x
Comparisons
Sometimes it is easier to show functions grow at the same
√ rate
by comparing
them
to
a
common
function.
Consider
x2 + 1
√
3
3
vs. 2x + 1. Differentiating would be a pain, so we wouldn’t
want to use L’Hopital’s Rule, but we could use something
simple, line x.
r
√
1
x2 + 1
x2
+ 2 =1
= lim
lim
2
x→ ∞
x→ ∞
x
x
x
√
3
lim
x→ ∞
2x3 + 1
=
x
Comparisons
Sometimes it is easier to show functions grow at the same
√ rate
by comparing
them
to
a
common
function.
Consider
x2 + 1
√
3
3
vs. 2x + 1. Differentiating would be a pain, so we wouldn’t
want to use L’Hopital’s Rule, but we could use something
simple, line x.
r
√
1
x2 + 1
x2
+ 2 =1
= lim
lim
2
x→ ∞
x→ ∞
x
x
x
√
3
lim
x→ ∞
2x3 + 1
= lim
x→ ∞
x
r
3
√
2x3
1
3
+ 3 = 2
3
x
x
Comparisons
Sometimes it is easier to show functions grow at the same
√ rate
by comparing
them
to
a
common
function.
Consider
x2 + 1
√
3
3
vs. 2x + 1. Differentiating would be a pain, so we wouldn’t
want to use L’Hopital’s Rule, but we could use something
simple, line x.
r
√
1
x2 + 1
x2
+ 2 =1
= lim
lim
2
x→ ∞
x→ ∞
x
x
x
√
3
lim
x→ ∞
2x3 + 1
= lim
x→ ∞
x
r
3
√
2x3
1
3
+ 3 = 2
3
x
x
Since they both
same rate as the same function, we
√grow at the√
3
2
conclude that x + 1 and 2x3 + 1 grow at the same rate.
Notation
Definition
A function f is of smaller order than g as x → ∞ if
lim
x→ ∞
f (x)
=0
g(x)
We indicate this by writing
f = o(g)
which is read “f is little oh of g”.
Notation
Definition
A function f is of smaller order than g as x → ∞ if
lim
x→ ∞
f (x)
=0
g(x)
We indicate this by writing
f = o(g)
which is read “f is little oh of g”.
Example
x = o(ex )
Notation
Definition
Let f (x) and g(x) be positive for sufficiently large x. Then f is
at most the order of g as x → ∞ if there is a positive integer M
for which
f (x)
≤M
g(x)
for x sufficiently large, We indicate this by writing
f = O(g)
which is read “f is big oh of g”.
Notation
Definition
Let f (x) and g(x) be positive for sufficiently large x. Then f is
at most the order of g as x → ∞ if there is a positive integer M
for which
f (x)
≤M
g(x)
for x sufficiently large, We indicate this by writing
f = O(g)
which is read “f is big oh of g”.
Example
x = O(ex ) because
x
ex
→ 0 as x → ∞, i.e. we can select an M.