Comparing growth rates? Determine which of the two functions grows faster, or state that they have comparable growth rates. 10xand ln x

Solution Step 1 In this problem we have to determine which of the two functions 10xand ln x, grows faster than the other otherwise we have to state the comparable growth rates. Suppose f and g are functions such that lim f(x) = , lim g(x) = x x g(x) Enough to prove: lim f(x)= 0which means f grows faster than g as x x l'Hôpital's Rule: f(x) 0 f(x) ± Suppose that we have one of the following cases, lim g(x)= 0 or lim g(x)= ± xa xa Where a can be any real number, infinity or negative infinity. f(x) (x) In these cases we have lim g(x)= lim g(x) xa xa