Expert: Scotto - 4/4/2010

Question
Hello, I will post the entire question and then explain my dilemma.

Suppose that f, g are positive functions. If lim as x-> infinity f(x)/g(x) = 0, we say that f grows more slowly that g or that g grows more rapidly than f. This may be indicated by writing:
f(x) << g(x). Arrange the following functions in order of rate of growth from least rapidly growing to most rapidly growing:

x
ln(x)
square root(x^3 + x^2 + x + 1)
e^x
ln(ln(x))
ln(e^x^2 + e^x + 1)
xln(x)
2^x^2
square root(10^x + 5^x +1)
x^x

For example I found out that for one of them I can do this:

2^x^2 >> x^x because,

ln(2^x^2) >> ln(x^2) therefore
x^2ln(2) >> xln(x)

I was thinking about trying to graph all of the equations and go from there but I am not certain that will yield anything. I was wondering if you can offer any advice as to how to go about solving this problem.

Answer
Take them one at a time and determine where they go.

Note that ln(x) < x^n < n^x < x^x for some constant n as x->∞.

To start with, take ln(x) << x << e^x.

The squareroot with the leading x^3 would be more than x, but less than e^x, so we have
ln(x) << x << √(x³+x²+x+1)<< e^x.

The ln(ln(x)) would be << ln(x), so no we have
ln(ln(x)) << ln(x) << x << √(x³+x²+x+1)<< e^x.

The ln(e^(x²)+e^x+1) would be around ln(e^x²) for large x, so it would be just above x².
Note that √(x³+x²+x+1) is just over x^1.5, so now we have
ln(ln(x)) << ln(x) << x << √(x³+x²+x+1) << ln(e^(x²)+e^x+1) << e^x.

Next note that xln(x) is >> than x and << than x^1.5, so we have
ln(ln(x)) << ln(x) << x << xln(x) << √(x³+x²+x+1) << ln(e^(x²)+e^x+1) << e^x.

Now for 2^x², that would be bigger than e^x, so we would have
ln(ln(x)) << ln(x) << x << xln(x) << √(x³+x²+x+1) << ln(e^(x²)+e^x+1) << e^x << 2^x².

For √(10^x + 5^x +1), that would approach 10^(x/2), which is greater than x to a power,
but less than e^x, so it would be ln(ln(x)) << ln(x) << x << xln(x) << √(x³+x²+x+1) <<
ln(e^(x²)+e^x+1) << e^x << √(10^x + 5^x +1) << 2^x².

Clearly x^x is the greatest, so we have ln(ln(x)) << ln(x) << x << xln(x) << √(x³+x²+x+1) <<
ln(e^(x²)+e^x+1) << e^x << √(10^x + 5^x +1) << 2^x << x^x.

You could check these in Excel by giving values for x and putting in each of the functions.
I believe that this is correct, but try each in Excel.
Not that on some of the functions near the end, x can't be too large.

On my PC, all of them can handle x=100, so try it out on yours and see what order they are at for large x.