Comparing growth rates? Determine which of the two functions grows faster, or state 1/2 1/3 that they have comparable growth rates. x and x

Solution Step 1 1/2 1/3 In this problem we have to determine which of the two functions x and x , grows faster than the other otherwise we have to state the comparable growth rates. Suppose f and gare functions such that limf(x) = , limg(x) = x x f(x) Enough to prove:lim x g(x)= which means f grows faster than gas x l'Hôpital's Rule: Suppose that we have one of the following cases, lim f(x)= or lim f(x)= ± xa g(x) 0 xa g(x) ± Where a can be any real number, infinity or negative infinity. f(x) f(x) In these cases we have lim g(x)= lim g(x) xa xa