Prove that lim+oo(lnVn)/ln2n =1/2

Question

And we have Un<n^1/2<Vn

And lim_+ooV_n=+oo

Answer 1

Hi there. Let's do this one!

lim ln(√n) ⁄ ln(2n) is of the indeterminate form ∞/∞ when n -> ∞, so we can use L'Hopital's Rule

n→∞

= lim 1/2 ln(n) / (ln2 + ln(n)) first simplifying, using the rules of logs.

n→∞

= 1/2 lim (1/n) / (1/n) differentiating top and bottom as per L'Hopital

n→∞

= 1/2 lim (1)

n→∞

= 1/2

Done!