lim (n to inf) [ ln (1+3n+4n^2) - ln(5+6n+2n^2) ]

Question

lim (n to inf) [ ln (1+3n+4n^2) - ln(5+6n+2n^2) ]

My Friend and I have trouble on this question

what I do is this:

lim (n to inf) [(1+3n+4n^2)/(5+6n+2n^2)] and then L'hopital's. the answer I got is 1

My friend write is like this:

ln lim(n to inf) [(1+3n+4n^2)/(5+6n+2n^2)]. she factor n^2. and the final answer is ln(2)

We did similar question at class : lim (n to inf) sin(1/sqrtn) --> sin lim (n to inf) (1/sqrtn)

We want to know what could be possibly wrong?

Answer 1

Hi,

If you factor an n² out of the numerator an denominator you have

Num: n² *(1/n² + 3/n +4)

Denom: n² *(5/x² + 6/n + 2)

when you form the ratio the n² 's cancel.

Now take the lim as n→∞ and all but the last term →0 and you are left with ln(4/2) =ln(2)

Does that make sense?

Jim