Which of the following functions grows the fastest as x goes to infinity

Question

a)2^x

b)ln(x)

c)sin(x)

d)x²⁰

^{my answer is d}

Answer 1

sin(x) never goes past 1, so that's immediately off the table. Just by looking at ln(x), we can already conclude that it's way slower than x^20 and 2^x. Comparing x^20 and 2^x, it might seem that x^20 is faster, and it is at the beginning, but at infinity, 2^x grows way faster. The easiest way to see this is to compare the growth rates. The derivative of x^20 is 20x^19, and 2^x is ln2(2^x). Comparing the growth rates of the growth rates, at some point of taking derivatives, the one coming from x^20 will become a constant, while the one coming from 2^x will still have an x. Working backwards shows us that, after a while, the growth rate increases faster for 2^x, which means 2^x grows faster as x approaches infinity. This is pretty much what L'Hopital's rule is.

This is kind of hard to visualize, but you can try playing around with it on a calculator. You'll see that at some point, 2^x starts to overtake x^20.