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)#

Explanation:

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