Examples from Section 6.7
Relative Rates of Growth

p. 503

Comparisons with (e^x)

1.

a) x + 3 grows more slowly than (e^x):

Limit[((x + 3)/(E^x)), x -> ∞]

b) (x^3) +  (sin^2) (x) grows more slowly than (e^x):

Limit[(((x^3) +  (sin^2) (x))/(E^x)), x -> ∞]

c) (Sqrt[x]) grows more slowly than (e^x):

Limit[((Sqrt[x])/(E^x)), x -> ∞]

d) (4^x) grows faster than (e^x):

Limit[((4^x)/(E^x)), x -> ∞]

e) (((3/2))^x) grows more slowly than (e^x):

Limit[((((3/2))^x)/(E^x)), x -> ∞]

f) (e^(x/2)) grows more slowly than (e^x):

Limit[((E^(x/2))/(E^x)), x -> ∞]

g) (e^( x))/2 grows at the same rate as (e^x):

Limit[(((E^x)/2)/(E^x)), x -> ∞]

h) (Subscript[log, 10]) (x) grows more slowly than (e^x):

Limit[Log[10, x]/(E^x), x -> ∞]

Comparisons with (x^2)

3.

a) (x^2)  +  4  x grows at the same rate as (x^2):

Limit[(((x^2)  +  4  x)/(x^2)), x -> ∞]

b) (x^5) - (x^2) grows faster (x^2):

Limit[(((x^5) - (x^2))/(x^2)), x -> ∞]

c) (Sqrt[1  +  (x^4)]) grows at the same rate as (x^2): ((Sqrt[1  +  (x^4)])/(x^2)) =  (Sqrt[(1/(x^4)) +  1]).

Limit[((Sqrt[1  +  (x^4)])/(x^2)), x -> ∞]

d) (((x + 3))^2) grows at the same rate as (x^2):

Limit[((((x + 3))^2)/(x^2)), x -> ∞]

e) x ln (x) grows more slowly than (x^2):

Limit[((x Log[x])/(x^2)), x -> ∞]

f) x (e^x) grows faster (x^2):

Limit[((x (E^x))/(x^2)), x -> ∞]

g) (e^(cos (x))) grows more slowly than (x^2):

Limit[(E^Cos)[x]/(x^2), x -> ∞]

h) (e^(x - 1)) grows faster (x^2):

Limit[((E^(x - 1))/(x^2)), x -> ∞]

Comparisons with ln (x)

5.

a) (Subscript[log, 3]) (x) grows at the same rate as ln (x):

Limit[Log[3, x]/Log[x], x -> ∞]

b) ln (2  x) grows at the same rate as ln (x):

Limit[Log[2  x]/Log[x], x -> ∞]

c) ln ((Sqrt[x])) grows at the same rate as ln (x):

Limit[Log[(Sqrt[x])]/Log[x], x -> ∞]

d) (Sqrt[x]) grows faster than ln (x):

Limit[((Sqrt[x])/Log)[x], x -> ∞]

e) x grows faster than ln (x):

Limit[(x/Log)[x], x -> ∞]

f) 5  ln (x) grows at the same rate as ln (x):

Limit[((5  Log[x])/Log)[x], x -> ∞]

g) 1/x grows more slowly than ln (x):

Limit[((1/x)/Log)[x], x -> ∞]

h) (e^x) grows faster than ln (x):

Limit[((E^x)/Log)[x], x -> ∞]

Ordering Functions by Growth Rates

7

(e^(x/2))grows more slowly than (e^x)as x -> ∞:

Limit[((E^(x/2))/(E^x)), x -> ∞]

(e^x) grows more slowly than ln (((x))^x):

Limit[((E^x)/Log)[x]^x, x -> ∞]

Rewriting:
(lim_(x -> ∞))((e^x)/(ln (((x))^x)))  = 
        (lim_(x -> ∞))(e
              ^(x (1  -  ln (ln (x))))) =  0
since

Limit[x (1 - Log[Log[x]]), x -> ∞]

ln (((x))^x) grows more slowly than (x^x):

\(Limit[Log[x]\^x\/x\^x, x -> ∞]\)

Rewriting:
(lim_(x -> ∞))((ln (((x))^x))/(x^x))  = 
        (lim_(x -> ∞))(e
              ^(x ( ln (ln (x)) - ln (x)))) =  0
since
(lim_(x -> ∞))x 
            (ln (ln (x)) - ln (x))  =  -∞