Help with Calculating Limit Problem

[gambix]

hi , i have a problem that i couldn't solve , i know its limit should be 1 because i looked up in the helping part of my manual .
i must calculate :

LIM (when n goes to infinit) ( (n^2 + n + 1) * ln( (n+1)/(n+2) ) * ln ( (2n+1)/(2n+3) ) )

i know i should use the case of 1 ^ infinit but i can't get it right .
thanks in advance for every answer

 

[MarkFL]

I have moved this thread to our Calculus sub-forum, because it appears to me that either L'Hôpital's Rule or a series expansion be used, which makes this a topic for the calculus.

I would try L'Hôpital's Rule myself. Can you rewrite the expression so that the limit is of the indeterminate form \(\displaystyle \frac{0}{0}\)?

 

[Opalg]

hi , i have a problem that i couldn't solve , i know its limit should be 1 because i looked up in the helping part of my manual .
i must calculate :

LIM (when n goes to infinit) ( (n^2 + n + 1) * ln( (n+1)/(n+2) ) * ln ( (2n+1)/(2n+3) ) )

i know i should use the case of 1 ^ infinit but i can't get it right .
thanks in advance for every answer

I would start by writing the limit as \(\displaystyle \lim_{n\to\infty}\Bigl(1+ \frac1n + \frac1{n^2}\Bigr) \Bigl(n\ln\frac{n+1}{n+2}\Bigr) \Bigl(n\ln\frac{2n+1}{2n+3}\Bigr)\) (dividing the first factor by $n^2$ and multiplying each of the other factors by $n$). If you can show that the limit of each of those three factors is $1$ then you can use the theorem that the limit of a product is the product of the limits.