Which of the following functions grows the fastest as x grows without bound?
f(x) = x10
g(x) = ln(x10)
h(x) = 10x
They all grow at the same rate.
yes i did mean that sorry
Thank you
To determine which function grows the fastest as x grows without bound, we need to compare the rates of growth of the three functions: f(x), g(x), and h(x).
Let's do a step-by-step analysis of each function:
1. f(x) = x^10: This function is a polynomial with the highest power of x equal to 10. As x grows without bound, the contribution of the other terms becomes negligible compared to the x^10 term. Thus, f(x) grows faster than any lower power of x.
2. g(x) = ln(x^10): This function is the natural logarithm of x^10. As x grows without bound, the logarithm of x also grows, but at a much slower rate compared to the exponentiation of x (as seen in f(x)). Therefore, g(x) grows slower than f(x).
3. h(x) = 10^x: This function is an exponential function with a constant base of 10. Exponential functions grow very rapidly as x increases, and the rate of growth increases as the base gets larger. Therefore, h(x) grows faster than f(x) and g(x).
In conclusion, as x grows without bound, the function h(x) = 10^x grows the fastest among the three given functions.
I will assume f(x) = x^10
also note that g(x) = ln(x^10) = 10lnx
Not sure if you know Calculus, so I will take an intuitive approach
Why not look at their graphs?
http://www.wolframalpha.com/input/?i=plot+y+%3D+x%5E10,+y+%3D+10ln(x)+,+y+%3D+10x
notice that h(x) is just a straight line and rises at a constant rate.
the ln function grows rather slowly, and is only defined for x > 0
the f(x) really takes off
So as x gets bigger , f(x) is the clear winner,
we could take a look at the race between the other two for some x
say, x = 5
f(5) = 5^10 = 50,000,000,000 , rather big
g(x) = 10ln5 = a mere 16.09..
h(x) = 50 , coming in 2nd place
x = 100
f(100) = 100^10 = 10^20 , really big
g(100) = 10ln(100) = 46.05..
h(x) = 500 , still running 2nd
x = 5000
f(5000) = not even worth doing, we know that's big
g(5000) = 85
h(x) = 50,000
looks like g(x) < h(x) < f(x)
Do you mean?
f(x) = x^10
g(x) = ln(x^10)
h(x) = 10x
They all grow at the same rate.
if so
if x = 1{1^10=1,ln 1^10=0 , 10*1 = 10}
if x = 10{10^10 =10billion, ln10^10 = 10ln 10 = 23 , 10*10 = 100
x^10 won hands down