Which of the following functions grows the fastest as x grows without bound?

Question

Which of the following functions grows the fastest as x grows without bound?

a)f(x) = x¹⁰

b)g(x) = ln(x¹⁰)

c)h(x) = 10^x

d)They all grow at the same rate

my answer is C

Answer 1

c) is correct as x grows without bound. exponentials>quadratics>logs, but not necessarily for specific intervals.

a) f(x) = x^10. f'(x) = 10x^9

b) g(x) = ln(x^10) = 10lnx. f'(x) = 10/1/x = 10x

c) h(x)= 10^x. f'(x) = (10^x)ln10)

from 0 to 1, a) grows from 0 to 1

b) grows from -infnity to 0

c) grows from 1 to 10

b) grows fastest on this particular interval, but c) ultimately grows fastest

from 1 to 2 a) grows from 1 to 1024

b) grows from 0 to 6.9

c) grows from 10 to 100

a) grows fastest on this interval 1 to 2

for larger intervals, c) grows fastest

x=100

100^10 = 1 x 10^12

ln(x^10) = 10ln100 = about 46

10^100 = 10^101 = 1 x 10^101

at larger intervals c) grows fastest. It's the best answer

compare the derivatives, the growth rates, 10x^9 = 10x

x^8 = 1

at x=1 or -1 the rates of increase are the same for a) and b)

10^x(ln10) = 10x

10^x(ln10) = 10x^9 solve for x to see where a) and c) are the same growth rate

and where b) and c) are the same

Answer 2

a) f(x) = x^10. f'(x) = 10x^9

b) g(x) = ln(x^10) = 10lnx. f'(x) = 10/1/x = 10x

c) h(x)= 10^x. f'(x) = (10^x)ln10)

from 0 to 1, a) grows from 0 to 1

b) grows from -infnity to 0

c) grows from 1 to 10

b) grows fastest on this particular interval, but c) ultimately grows fastest

from 1 to 2 a) grows from 1 to 1024

b) grows from 0 to 6.9

c) grows from 10 to 100

a) grows fastest on this interval 1 to 2

for larger intervals, c) grows fastest

x=100

100^10 = 1 x 10^12

ln(x^10) = 10ln100 = about 46

10^100 = 10^101 = 1 x 10^101

at larger intervals c) grows fastest. It's the best answer

compare the derivatives, the growth rates, 10x^9 = 10x

x^8 = 1

at x=1 or -1 the rates of increase are the same for a) and b)

10^x(ln10) = 10x

10^x(ln10) = 10x^9 solve for x to see where a) and c) are the same growth rate

and where b) and c) are the same