Question

pls help with this...

Answer 1

\[\lim_{n \rightarrow + \infty} \frac{ n \times \ln n}{ (n+1)\times \ln (n+1) }\]

Answer 2

can i use l'hospital rule?

Answer 3

\begin{align}&\lim_{n \rightarrow + \infty} \frac{ n \times \ln n}{ (n+1)\times \ln (n+1) }=\\
&\lim_{n\rightarrow + \infty} \frac{n}{n+1}\times \lim_{n\rightarrow + \infty}\frac{\ln n}{\ln(n+1)}=\\
&1\times 1=\\
&1
\end{align}

Answer 4

so, l'hospital rule?

Answer 5

how did u get 2nd limit as 1??

Answer 6

No, that's just using the product rule. L'hospital's rule would be to differentiate the top and bottom...which you don't really want to do.

Answer 7

As n gets very large, \(\ln n\) gets very close to \(\ln(n+1)\).

Answer 8

thats logic,but how will u prove that mathematically?

Answer 9

Actually, I stand corrected, you do prove that part with L'hospital's rule:
\(\displaystyle\lim_{n\rightarrow\infty} \frac{\ln n }{\ln(n+1)} = \lim_{n\rightarrow \infty} \frac{\frac{1}{n}}{\frac{1}{n+1}}=\lim_{n\rightarrow \infty}\frac{n+1}{n}=\lim_{n\rightarrow\infty}\frac{1}{1}=1\)

Answer 10

:)

Answer 11

there u go,thanks :)

Answer 12

thx for your help!