Put limit in big O little O notation?

So for the two functions, lnx and x-lnx. Compare which one grows faster and put it in the big o little o notation ( slower = o (faster) , top = O (bottom))

1 Answer
Jul 13, 2018

lnx = o(x-lnx)lnx=o(xlnx)

Explanation:

Supposing we are here required to evaluate the behavior for x->oox, by definition, given two real functions f(x)f(x) and g(x)g(x):

lim_(x->oo) (f(x))/(g(x)) = 0 <=> f(x) = o (g(x))

and:

lim _(x->oo) "sup"abs((f(x))/(g(x))) < oo <=> f(x) = O(g(x))

In our case let f(x) =lnx and g(x) =x-lnx. Then:

lim_(x->oo) (f(x))/(g(x)) =lim_(x->oo) lnx/(x-lnx)

lim_(x->oo) (f(x))/(g(x)) =lim_(x->oo) 1/(x/lnx-1)

Evaluate now the limit:

lim_(x->oo) x/lnx

it is in the indeterminate form oo/oo so we can use l'Hospital's rule:

lim_(x->oo) x/lnx = lim_(x->oo) (d/dx (x))/(d/dx (lnx))

lim_(x->oo) x/lnx = lim_(x->oo) 1/(1/x) = lim_(x->oo) x = +oo

Then:

lim_(x->oo) 1/(x/lnx-1) = 0

which means:

lnx = o(x-lnx)