9. Relative Rates of Growth
P. K. Lamm
Lecture Notes:
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9. Relative Rates of Growth
These classnotes are intended to be supplementary to the textbook and are necessarily limited
by the time allotted for classes. For full and precise statements of definitions and theorems,
as well as material covering other topics and examples, please consult the textbook.
1. Exponential growth.
If you read the pages under the topic “exponential growth” in Wikipedia, you will find the
following story:
When the creator of the game of chess showed his invention to the ruler of
the country, the ruler was so pleased that he gave the inventor the right to
name his prize for the invention.
The man, who was very wise, asked the king this: that for the first square of
the chess board, he would receive one grain of wheat, two for the second one,
four on the third one, and so forth, doubling the amount each time. The
ruler, who was not strong in math, quickly accepted the inventor’s offer,
even getting offended by his perceived notion that the inventor was asking
for such a low price, and ordered the treasurer to count and hand over the
wheat to the inventor.
However, when the treasurer took more than a week to calculate the amount
of wheat, the ruler asked him for a reason for his tardiness. The treasurer
then gave him the result of his calculation, and explained that it would be
impossible to give the inventor the reward.
Why?
• The inventor wants 2n−1 grains of wheat on the nth square, for each n = 1, . . . , 64.
• On the 21st square: over 1, 000, 000 grains of wheat
• On the 41st square: over 1, 000, 000, 000, 000 (trillion) grains of wheat
• Total amount due the inventor:
– approximately 80 times what could be produced in one harvest (in modern times) if all
of the Earth’s land were used to grow wheat; or,
– over 8 cubic miles of wheat in volume.
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We already know that
lim ax = ∞
x→∞
for any a > 1, but the point to get from this story is that this growth as x → ∞ is extremely fast.
On the other hand, logarithmic functions also grow without bound as x → ∞, i.e., for a > 0,
ln x = 1
lim ln x = ∞,
lim | loga x| = lim x→∞
x→∞ ln a | ln a| x→∞
but as we shall see below, logarithmic functions grow very slowly.
What about polynomials? We know that
lim xn = ∞,
x→∞
for n ∈ (0, ∞). But for some a > 0, does xn grow faster or slower than ax as x → ∞? Faster or
slower than loga x, for a > 0?
2. Relative rates of growth of functions
Definition: Suppose f and g are positive functions for x sufficiently large, and that
lim f (x) = ∞,
lim g(x) = ∞.
x→∞
x→∞
Then we say f grows faster than g as x → ∞, or g grows slower than f as x → ∞, if
f (x)
= ∞,
x→∞ g(x)
lim
or, equivalently,
g(x)
= 0.
x→∞ f (x)
lim
We say that f and g grow at the same rate as x → ∞ if
f (x)
= L,
x→∞ g(x)
lim
for L > 0 finite.
Example 2.1: To compare the growth as x → ∞ of the functions ln x and x using the above
definition, we consider
ln x
lim
x→∞ x
∞
which requires l’Hôpital’s rule because it is a limit of the form
. We then have
∞
ln x
1/x
= lim
= 0,
x→∞ x
x→∞ 1
lim
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so it follows that ln x grows more slowly than x as x → ∞.
Example 2.2: Next we compare ex to x3 . Using l’Hôpital’s rule (repeatedly) we have
ex
ex
ex
ex
=
lim
= ∞,
=
lim
=
lim
x→∞ 6
x→∞ 3x2
x→∞ 6x
x→∞ x3
lim
Thus ex grows faster than x3 as x → ∞.
Example 2.3: To take the last example further, we’ll compare ex to |p(x)| for any polynomial p
of degree n > 0.
Assume p has exactly degree n for some n ≥ 1, i.e.,
p(x) = an xn + an−1 xn−1 + · · · + a1 x + a0 ,
for a0 , a1 , . . . , an ∈ R, an 6= 0. Then
p(n) (x) = n · (n − 1) · · · 2 · 1 · an = n! an ,
but for k = 1, . . . , n − 1,
lim |p(k) (x)| = ∞,
k→∞
since p(k) is a polynomial of degree greater than zero. We must then use l’Hôpital’s rule n times
to get
n! |an |
|p(x)|
= lim
= 0.
lim
x
x→∞
x→∞
e
ex
So ex grows faster as x → ∞ than |p(x)| where p is any polynomial.
Example 2.4: What about a comparison of the growth as x → ∞ for two polynomials?
Assume that pn and pm are polynomials of exactly degree n and m, respectively, for n > m > 0.
|pn (x)|
∞
pn (x)
Since lim
is a limit of the form
, we repeatedly use l’Hôpital’s rule on lim
x→∞ |pm (x)|
x→∞ pm (x)
∞
until the polynomial in the denominator is reduced to a nonzero constant (after m differentiations).
Then the resulting polynomial in the numerator is of degree n − m > 0 which goes to ∞ in
magnitude as x → ∞. Thus
(m)
|pn (x)|
lim (m)
= ∞,
x→∞ |p
m (x)|
and it follows that an increase in the degree of a polynomial will result in a function which grows
faster (in magnitude) as x → ∞.
Example 2.5: If we compare ax to bx , for a > 1, b > 1, a 6= b, then (assuming a > b > 1),
a x
ax
lim x = lim
= ∞,
x→∞ b
x→∞ b
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so an increase in the size of the base leads to an increase in the rate of growth of the function as
x → ∞.
Example 2.6: Given the results in Example 2.4 above, one might expect an analogous finding
when one changes the base a > 0 for loga x. In fact, if a > b > 1, we then have
lim
x→∞
ln x/ ln a
ln b
loga x
= lim
= lim
= L,
x→∞
x→∞
logb x
ln x/ ln b
ln a
where L is finite. We get a similar result for 1 > a > b > 0 provided we compare | loga x| to
| logb x| in that case. Thus, in contrast to exponential functions, all logarithmic functions grow at
the same (slow) rate as x → ∞.
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